(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y

Transcription

1 [ ] y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101) y d2 y 2 = ( ) 2 1 p = p y 1 (2) 1 ( 18) ( s180101) 0.5 = 3x 4y = x 2y ( ) ( ) ( u 1 x(t) = exp(λt) u 2 y(t) ( ) ( ) λ u 1 u 2 (2) t = 0 u 1 u 2 u 1 + u 2 = 1 ( 0.6 x(t) y(t) ) ( 6 3 ) u 1 u 2 ( 19) ( s190101) y = ax (a 0) 1,. (2) 1., C C (x, y), (x, y) C C. ) 1

2 (3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y 2 ) 2xy = 0 (3) (2), D x y y =, y = d2 y 2. y 3y = e x (2) y + 2y + y = ( 21) ( s210104) ( 22) ( s220101) ( x d y 1 y 2 ) = ( ) = ( y 1 y 2 ) ( 23) ( s230101) = x (A),.. (A) y = a 3 x 3 + a 2 x 2 + a 1 x, a 1, a 2, a 3. (2) (A). ( 24) ( s240103) 0.11,.,., y(x) y = 0 (2), y(x)., n y = cos nx 2 2

3 (3) g(x), 2π, 1.. g(x) = π 2 8 π 2 8 ( 1 + 2x ) ( π 1 2x ) π ( π x < 0) (0 x < π) (4), y(x)., (3) g(x). + 10y = g(x) 2 ( 25) ( s250102) 0.12.,. (2x 2 + y 2 ) = 2xy (2) y = 0 ( 26) ( s260101) 0.13.,. ( ) = 2 y 2 + y (2) y = 9e x ( 28) ( s280101) y = xy x y + 1 (2) y 6y + 5y = 13 cos x 0.15 r = f(θ) d 2 f dθ 2 + f = a a (2) ( 29) ( s290107) θ = 0 f = 2a, df dθ = 0 (3) θ π θ π (r, θ) r = f(θ) (4) ( 8) ( s080302) 0.16 y = C 1 e Ax + C 2 e Bx C 1 C 2 y 5y + 4y = 0 (2) y = Ae Bx y 5y + 4y = e x 3

4 (3) y + P (x)y + Q(x)y = R 1 (x), y + P (x)y + Q(x)y = R 2 (x) y = f(x), y = g(x) y = f(x) + g(x) y + P (x)y + Q(x)y = R 1 (x) + R 2 (x) (4) (3) y 5y + 4y = e x + e 4x ( 9) ( s090303) 0.17 dq(t) = q(t) CR, q(t).,c, R, q(t) t = 0 q(0) = q 0 q 0. ( 16) ( s160305) 0.18, y = e λx y = 0 ( 16) ( s160306) 0.19 (2x y + 1) (x 2y + 5) = 0,. 2x y + 1 = 0, x 2y + 5 = 0. (2) (X, Y ),. (3) (2). (4) (3), (x, y) , y =, y = d2 y 2. ( 18) ( s180303) y = y 2 + y (2) y + 2xy = 0 (3) y 4y + 3y = x ,. (x 1) + 2y = 0 1 ( ) 2 (y 1) d2 y = 0 2 ( 20) ( s200303) 1. (2) 2, = u, y u 1., 2 = du u. (3) (2) 1. 4

5 (4) 2. ( 21) ( s210305) d 2 x 2 + ω2 x = sin Ωt,.,., ω Ω. ω Ω,. (2) ω = Ω,. t = 0, x = = 0 ( 22) ( s220303) 0.23 a 2 + b 2 = 5, a > 0, b < 0,. (axy e x cos y) = ( e x sin y + by 2) a b. (2), y + 9y = 0,. ( 23) ( s230304) y = A sin 3x B cos 3x A, B. (2) x = 0 y = 1, y = 3. (3) x = π 3 y = 1, x = π y = y + 2y + 2y = 85 sin 3x,. ( 24) ( s240304) y = 6 cos 3x + 7 sin 3x 1. (2). (3) x = 0 y = 0, y = ,. = x + sin t = x + 2y x(t), y(t). ( 25) ( s250304) (2) y = A sin t + B cos t, A, B. (3) (2),. ( 26) ( s260304) 5

6 0.27 d2 x 2 + ω2 x = 0., ω 0. (2) ω = 1, d2 x 2 + ω2 x = 2 sin 3t. (3) (2), t = 0 x = 0, = y + 3y + 2y = 4x 2 14,. S 2 + 3S + 2 = 0. (2) y + 3y + 2y = 4x ( 27) ( s270304) (3) (2), y(0) = 1, y (0) = = 2 2x + y 1. ( 28) ( s280303) u = 2x + y 1,. (2). (3) x = 1 y = xy = y(y 1) 0.31 y = f(x) ( 29) ( s290304) ( 13) ( s130407) P (a, b) x ( 1 2 (a+b2 ), 0) y = f(x) (2) (0, 2) 3 x 1 e = 2.718, e 1 = 0.367, e 1.5 = ( 5) ( s050503) 0.32 f(x) > 0 [a, b] f (x) x A(a, 0) P (x, 0) a < x b A, P x y = f(x) B, Q BQ (2) y = f(x), x AB, P Q BQ k ( 6) ( s060502) 0.33 y = f(x) P x N P N (2) P N P y k (0, 1/k) ( 7) ( s070502) 6

7 0.34 y(0) = a (2) x y(x) = xy = x x3 ( ) e t2 (cos xt + x 2 t) ( ) ( 8) ( s080502) 0.35 f(x) x 2 d2 f(x) 2 2x df(x) + 2f(x) = 0 (x 1) (a) df x = 1 f = 1, = 0 (b) (5) (a) t = log x (t) 2 3 (t) + 2y(t) = 0 (b) (c) t = 0 y = 1, = 0 (d) (2) (c) y(t) = C 1 e λ1t + C 2 e λ2t (c) (d) λ 1, λ 2, C 1, C 2 (3) (b) (a) (4) z(t) = (t) 2 2 A (c) (t) ( ) y(t) dz(t) = A A z(t) (5) A (2) λ 1, λ x, f(x) f (x) f(x) = 0 (e) ( 15) ( s150502), f(0) = 0, f (0) = 1., f(x) g(x). g(x) = f(x). (2) g g( 1). (3) h(x)., h. x 0 tf(t) h(x) = d x 2 g(t) x 7

8 0.37 y = y(x) (y 0), z = z(x).,. z = y 4, dz y. ( 17) ( s170503) (2) z = y 4, + yp (x) = y5 Q(x) z. (3) + xy = 1 2 xy5. ( 19) ( s190506) 0.38 t, x = x(t), y = y(t) xy P (x(t), y(t)). x(t) y(t) = αx y = x + αy (x(0), y(0)) = (1, 1)., α.. α = 0, x(t) y(t). (2) α 0, x(t) y(t). (3) t (t 0), P α > 0, α = 0, α < (t) λ y(0) = a ( 20) ( s200502) + λy(t) = 0 (i) (2) (t) + λy(t) = f(t) (ii) f(t) y(0) = a y(t) = e λt x(t) (iii) (ii) x(t) x(t) y(t) ( 9) ( s090606) 0.40 d 2 u 2 + ω2 u = 0 12 ( s120609) 0.41 x = x(t) d 2 x 2 + a + bx = 0 a b a 2 > 4b 8

9 x 1 (t) x 2 (t) c 1, c 2 x 3 (t) = c 1 x 1 (t) + c 2 x 2 (t) (2) α x(t) = e αt p(x) 2 + q(x)y = 0 y 1 (x), y 2 (x), d 2 z dz + p(x) + q(x)z = f(x) 2 ( 15) ( s150609). z(x) = c 1 (x)y 1 (x) + c 2 (x)y 2 (x) dc 1 y 1 + dc 2 y 2 = 0 c 1 (x), c 2 (x), z(x) = y 1 (x) x f(x )y 2 (x ) W (x ) x f(x )y 1 (x ) + y 2 (x) W (x ) 2., W = y 1 y 1 2. ( 21) ( s210609) 0.43 d 2 x 2 = a2 x 3. a. (2) 3 at << 1, d2 x 2 = 0. (3) 3 t = 0 x = x 0, / = v 0, t. ( 22) ( s220607) 0.44 m k, x., x 1, x 2.. m d2 x 1 2 = k(x 1 x 2 ), m d2 x 2 2 = k(x 2 x 1 ). (2) X = (x 1 + x 2 )/2 x = x 2 x 1,. (3). (4).. ( 24) ( s240604) 0.45 dn = λn λ. t = 0 N = N 0. ( 28) ( s280605) 9

10 0.46 d2 x x = 0,. ( 28) ( s280606) (x + 3y) + (3x y) = 0 (2) y + 2y 3y = e x 0.48 ( ). ( 28) ( s280613) (2) = xy (x = 0, y = 2) y = 2x (x = 0, y = 5 x = 1, y = 6) ( 29) ( s290608) 0.49 x(t) + a t f(t) x(t) = f(t) + b t 0 0 f(t) a > b > 0, lim x(t) = 1, t + 0 lim x(t) = 0 t 0 x(t) f(t) (2) x(t) 1 O t f(t) = 1 t 0 f(t) = 0 t < 0 ( 9 ( s090701) 0.50 d 2 x 2 + 2b + ω2 x = 0 b 2 ω 2 0 (2) t = 0 x = 0 = 1 b > f(θ) = n=0 n=0 π n ( n n! sin ) π 4 θ n ( n n! sin ) π 4 f (θ) ( 10 ( s100701) (2) f (θ) f(θ) (3) f(θ) f(π) = n=0 π n n! sin ( n ( 11 ( s110702) ) π 10

11 0.52 t = 0 x = 1, y = = x + 2 = y 0.53 ( 12) ( s120703) 3x y = (x + y) (2) 6x 2y 3 + ( 2x 2y + 1) = 0 ( 15) ( s150702) 0.54 x 2 y 2xy + 2y = 0 (2) y + Ay + By Cx D = 0 ( A, B, C, D ) (3) y (3x + 2y 2 ) = 0 (4) yy + (y ) 2 5y = 0 y =, y = d2 y 2 ( 16) ( s160703) (a) (2xy + x 2 )y = 2(xy + y 2 ) (b) y + 2y cos x = sin(2x) (2). y + y 2y = 3e x, y(0) = 1, y (0) = 1 ( 17) ( s170701) y = f(x) 2 (a) x = 0 y = 1 = 0 f(x) = 0 (2) f(x) = sin 2x (a) y p = x(a cos 2x + B sin 2x) A, B (3) f(x) = 100 N=1 sin Nx y S x y S x x ( 18) ( s180703) xy = x2 + 2y 2 (2) d2 y 2 + n x + a2 y = 0,., n, a 0. (a) n = 0. (b) n = 2. 11

12 ( 19) ( s190701) 0.58 P (x, y) + Q(x, y) = 0, u(x, y) du(x, y) = u u +, x y u(x, y) = C (C )... (a) y + x = 0,, 1 xy α (α ). α,. (b) (a), x = 1 y = e., e. (2) (a) x 2 d2 y 2 x + y = 0 (2), x = e t., e. (b) (a) (2). (c) x 2 d2 y 2 x + y = x log e x, x = 1 y = 1, 0.59., y =, y = d2 y 2.. (2) y + 2y 3y = x + cos x = 0. ( 21) ( s210705) 2yy 3(y ) 2 = y 2 y(0) = 1, y (0) = 1 ( )., y > 1/2, y > 0. (a) p = y, ( ) p y 1 f(y, p) dp 3p2 = y 2. f(y, p). (b) ( ), p y. (c) ( ), y x. ( ) ( 22) ( s220702) 0.60., y =, y = d2 y 2. y = p(x) + q(x)y + r(x)y 2, y 1 (x). ( ) 12

13 (a) ( ) y = y 1 (x)+1/u(x), u(x) p(x), q(x), r(x), y 1 (x). (b) y = (x 2 + x + 1) (2x + 1)y + y 2, y 1 (x) = x.. (a). (2) αy + y + y = 0 y(x = 0) = 1, y (x = 0) = 2 ( ). (a) y = C 1 e λ1x + C 2 e λ2x. λ 1, λ 2, C 1, C 2 α., e. (b) x 0, ( ) y + y = 0 y(x = 0) = β. α, β, (a) λ 1, λ 2, C 1, C 2. ( 23) ( s230701) 0.61 = (y k)y ( )., k. (a) y k, ( ). (b) x = 0, y = y 0. ( )., y 0 > 0. (c) (b), y 0 k, y x. (2)., e. + 2y = sin x + e 5x ( 24) ( s240701) 0.62, y(0) = 0, y (0) = 1., e,, y =, y = d2 y 2. y 4y + 3y = e x ,. ( 25) ( s250706) f (2) (x) 2αf (x) + α 2 f(x) = 0 ( ) f (n) (x) f(x) n n, α 0.. x, k e kx, x 3., e. (2) f(x), ( ) n,. f (n+2) (x) 2αf (n+1) (x) + α 2 f (n) (x) = 0 f (n) (x), f (x) f(x). 13

14 (3) f(x) f(x) = f (m) (0) xm m!, (2) f (n) (x) m=0 [ ], f(x) = f(0)e αx + f (0) αf(0) xe αx., m 0, f (0) (x) f(x), 0! = 1. (4), f(0) = 1, f (0) = p 2 p. f (2) (x) + 4f (x) + 4f(x) = e 2x (5) (4) f(x), f(x) = 0 p, p f(x) y(0) = tan x 2 y = 1 x e + 1 ( 26) ( s260701) (2) π 4 < x < π 4 y 2 + 4y = 1 cos 2x. (a) 2 + 4y = 0 y 1, y 2, W (y 1, y 2 ). ( ). (b) ( ), u(x), v(x) 2 W (y 1, y 2 ) = y 1 y 1 2 du y 1 + dv y 2 = 0 y = u(x)y 1 + v(x)y 2, u(x), v(x) 1. (c) ( ) ;, y =, y = d2 y 2. y (a + 2)y + 2ay = f(x)., a. (a) f(x) = 0,. (b) f(x) = 5e 3x a < 0,., e. ( 27) ( s270701) 14

15 (2). x 2 y 4xy + 6y = 0 (3), y 1 (0) = 4, y 2 (0) = 3. { y 1 + 2y 2 = 2y 1 + 5y 2 2y 1 y 2 = 14y 1 + 5y y(x). = y2 + 2xy x 2 2x 2 ( 28) ( s280701) (2) q(t),., R, C, E 0., q(t) CE. (a). R dq + 1 C q = E (b) t = 0, q(t) = 0. (c) (b) t 0. (3) x(t) 2, m d2 x 2 + c + kx = 0 1, A. [ ] [ ] d x 1 x 1 = A x 1 = x, x 2 =, m, c, k, m x 2 x y = ex y(0) = 2, y (0) = 0., y =. (2) = (1 y)y., y(0) = 1 2. (3) x > 0 u(x) d 2 u 2 + A x du u = 0 ( ),., A A 1. ( 29) ( s290701) 15

16 (a) d 2 f α x d f α (1 + α2 x 2 ) f α = 0 f α (x) 0., α., ( ), g(x) u(x) = f α (x)g(x), [ ] [ ] d f α ( ) + ( ) f α = 0. ( ), ( ), g(x), A, α. (b) (a) ( ), ( ), g(x) α A.,, C. ( 30) ( s300701) 0.68 x 1 + y 2 + y 1 + x 2 = ( ) = y + xy2 z(x) = 1 y(x) (2) ( ) ( 8 ( s080803) ( 10 ( s100802) 0.70 y 2y + 5y = e x y(0) = p, y (0) = q ( 11 ( s110802) 0.71 y + 2y + y = e x y(0) = 1, y = e/4 ( 15) ( s150802) 0.72 t x(t). d 2 x 2 = t, x(t). (2) t +, x(t) +, x(t) y + y = 0. (2) y + y = e x t x(t) ( ),., c. d 2 x cx = 0 ( 21) ( s210804) ( 26) ( s260804) 16

17 x(0) = 0 0 t 1 x(t) 0 0 ( ) x(t) c. (2) c, ( ) ( ) x(0) = x = 0, (0) = 1, c. ( ) ( ) y 3y + 2y = 0 (2) y 3y + 2y = e x 0.76 y = y x y = y(x) y = e π d2 y x y,. ( 27) ( s270804) ( 8) ( s080905) ( 18) ( s180907) y = 0 y = y(x) y(2) = 3 lim y(x) = 0 x + ( 19) ( s190901) y + y = 1. ( (2) 2y y = y 3 x = 0, y = 1 ), z = 1, z 2 y x y = y(x) y 4y + 4y = 4x, y(0) = 0, y = y = cos 2x y = y(x). (2) y = y(x) lim x + + 3y = 1 y(x). ( 20) ( s200904) ( 22) ( s220904) ( 23) ( s230901) 0.81 y 2y + y = (3x + 1)e x y = y(x), y(0) = 3, y (0) = 2. ( 24) ( s240904) 17

18 0.82 x y = y(x). y = y(1 y), y(0) = 1 3. (2) y + 4y = e x, y(0) = 6 ( π ) 5, y = e π 4. ( 25) ( s250905) 0.83 y + 4y + 4y = e 2x 1 + x 2 y = y(x) y(0) = 0, y (0) = d2 y 2 + 3y = cos 3x y = y(x), y(0) = 1, x y = y(x) y y 6y = 6x + 4e x, y(0) = 0, y (0) = 0. ( 26) ( s260903) (0) = 1, y ( 27) ( s270904) ( 28) ( s280904) 0.86 y + 2y + 5y = 10 cos x y = y(x), y(0) = 0, y (0) = 1, y. y =, y = d2 y 2. ( 29) ( s290904) 0.87 R α. t x. k(> 0). x t. (2) A, x. (3) T. (4) R α, ( 17) ( s171008) sin x cos 2 y cos2 x = 0 (2) + y tan x = sin 2x (3) y = sin 2x ( 21) ( s211004) 0.89 d 2 x 2 = f(x) x 1 f(x) = 1 1 < x < 1 f(x) = x x 1 f(x) = 1 18

19 t = 0 x = 1 = 0 (2) t = 0 x = 3 2 (3) t = 0 x = 5 2 = 0 0 t 2 = 0 ( 4 ( s041101) 0.90 = x 2xy = y + xy x-y (2) t = 0 x = 1, y = 2 x y (3) (2) x y (4) t = 0 x, y x y t ( 5 ( s051101) 0.91 xy P (x, y) t 0 t = 0 x = 1, y = 3 = y = x ( x > 0, y > 0 ) x y t (2) P x = 0 y t (3) ( x < 0, y > 0 ) x y t P (4) P x x = 1 ( 6 ( s061101) λ2 y = f(x) ( ) ( λ 0, 0 x l ) y(0) = y(l) = 0 y = y(x) ( ) 0 λ (2) λ (*) ( 7 ( s071101) 0.93 = x + x2 t = 0 x = ξ x = ϕ(t, ξ) x = ϕ(t, ξ) (2) t x = ϕ(t, ξ) ξ ξ (3) ξ x = ϕ(t, ξ) t t 19

20 (4) x = ϕ(t, ξ) < t < t ξ ( 8 ( s081101) 0.94 xy P (x, y) t 0 = x y = x + y (2) t = 0 x = 1, y = 3 x y t (3) x = 0 y t (4) P ( 9 ( s091101) λx = f(t) t = 0 x = a λ (2) f(t) sin t (3) ( 10 ( s101101) 0.96 a f(x) a 2 < x < a 2 f(x) = 1 a x < a 2 x > a f(x) = 0 2 x ± a 2 d2 y 2 y = f(x) x = ± a 2 x ± y = 0 y(x) y( x) = y(x) (2) (3) a ( ) y = x(x y) ( 13) ( s131101) x = 0 y = 0 y(x) x 0 ( 15) ( s151101) 20

21 0.98. = x 1 2x 2 y ( 17) ( s171103) y = 6x3 + 3x 2 14x + 3 (2) + 2xy = x y 2 ( 18) ( s181101) 0.100,. (2x + y) + (4x + 2y 3) = 0 (2) 2 6y = 4e2x : x = 0, y = 2, : x = 2, y = 1 = 0 ( 19) ( s191102) = x + y (2) = y2 x 2 2xy ( 20) ( s201102) y + P (x)y = Q(x) 1..,. 1. y + P (x)y = ( ),,, C,. y = C ( ) (.), 1., 1 y = C(x) ( ) 2. 1 y + P (x)y = C ( ) = Q(x), C = Q(x) ( ) C(x), 2, 1. y = e R [ R ] P (x) P Q(x)e (x) + C (2),,. 21

22 (3). y y = e x x 2 y = 0 ( 20) ( s201103) (2) x + y + 4x = 0 ( 21) ( s211101) = 2y5 6x 4 y 3xy 4 3x 5 (2) 2xy = 2x ( 22) ( s221102) x 2 = y2 + xy x 2 (2) ( xy 3 + 3x 3 y ) ( y 4 + 4x 2 y 2 + x 4) = 0 + y = cos x ( 23) ( s231102) (2) xy = 4 y 2 ( 24) ( s241102) x y = x 2 + y 2 (2) x = 0 ( 25) ( s251102) y = 3e 2x (2) (x + 1) = (2x + 3)y ( 26) ( s261102) 0.109, x(x 2 1) + 2y(x2 x 1) = 0 (2) = 2xy x 2 + y 2 ( 27) ( s271102) 22

23 = y + ex sin x (2) x 1 + y 2 + y 1 + x 2 = 0 ( 28) ( s281102) y = 4x 2y 2x y 1 (2). y y + y = 0 (3) xy y x log x = 0, y = 0. (2) + Kx = 1 [0, L]. 2. d 3 y 3 = xex. ( 28) ( s281103), y(0) = y(l) = 0 (3) (1 + exp(x)) = y. y(0) = 1. ( 29) ( s291101) = (y x)2 (2) = (y + cos x) sin x ( 29) ( s291104) = x xy ( t > 0 ) = y + xy x(0) = a y(0) = b (a, b) I ( x(t), y(t) ) I y + 4y + 4y = x 2 d 2 x 2 + 2a + b2 x = 0 ( a < b ) ( 6 ( s061202) ( 7) ( s071203) ( 8 ( s081203) d 2 x + x = sin 2t (t > 0), 2 x(0) = 0, (0) = 0 ( 9 ( s091202) 23

24 0.118 y = y(x) d y(x) + 2y(x) = x, y(0) = 1 ( ) w(x) = e 2x y(x) w ( 10 ( s101203) = 5x y = x 3y ( 11 ( s111203) x 2 d2 y + 4x 2 + 2y = 1, y(x) x=1 = = 0 x=1 (i) x = e t x [1, ] t (2) y(x(t)) = y(x(t)) 2 d2 y 2, d 2 x 2,, (3) x = e t (i) y = 1, y(t) t=0 = = 0 (ii) t=0 (4) (ii) (5) (4) (ii) (i) y(x) ( 13) ( s131202) x y = 2 cos x : y(0) = 1, y (0) = = 3x y + 1 = x + 3y = xy = xy + 2 /2 xe x (2) y(0) = 1 ( 14) ( s141203) ( 15) ( s151204) ( 16) ( s161203) 24

25 0.124 t (x(t), y(t)) ( ) ( ) d x(t) αy(t) + b = y(t) αx(t) + a., a, b α(> 0). ( x(t) y(t) ) = A(t) ( x(0) y(0) ) (A(t) I) v., A(t) t 2 2, I 2 2, v (x(0), y(0)).. (t) t. (t) (2). (3) A(t). (4) (x(t), y(t)) (2) ( 17) ( s171204) = a(y + b), a, b, y(0) = y 0 (y 0 b). = ky(p y), k, p, y(0) = y 0 (0 < y 0 < p). ( 19) ( s191209) f(x) < x <,. f(x) = 1 + x 0 (t x)f(t) f(0), f (0). f(x). (2) f(x) , x x x x 2 y = 0 y = x 2. ( 20) ( s201204) (2) y = ux 2 u x, u. (3) (2) u,. ( 21) ( s211204) 0.128, x(t), y(t). 3x + y = 0 x + y = 0 ( 22) ( s221204) 25

26 (2) = 2xy + 3x = 3xy 5xy 1 3 y 4/3 z z y + y + y 2 = 0, y(0) = 1 (2) y + y + x = 0, y(0) = ( 23) ( s231204) ( 24) ( s241203) (2) 2 + 2y = 8e2x = 2x y ( u(x) = y ) x 2y x ( 25) ( s251204) d 2 x x = 0, x(0) = 1, x (0) = = 0, t 0 t=0 (2) x(t) y(t) =, O xy P ( x(t), y(t) ). t 0 P (x, y). P. ( 26) ( s261204) 0.133,. ( x 2 xy ) + y2 = 0 x = e, y = e,, e. y = u y(x) u(x) x (2) y = 0, x 0 x = 0, y = 1, y = 1 ( 27) ( s271204) (2) + 2y = sin t 2 t = 0, y = 1, = xy = 4xex. ( 28) ( s281204) 0.135, x f(x)., f(0) = 1, f (0) = 0. ( d 2 ) 2 ( f(x) d 2 ) f(x) = 0 ( 28) ( s281205) 26

27 (2) 2 + 2y = 2, t = 0 y = 2, = y = cos t 2t. ( 29) ( s291204) x 4 + y2 = 0 ( 16) ( s161316) y t y(t) 3. y(t). (2) t = 0 y(t) 1, t = 1 y(t). ( 16) ( s161317) (x) m = mx ( 1) ( 1) t = 0 x = x 0 (2) (x) (K x) = mx(k x) ( 2) K ( 2) t = 0 x = x 0 x x ( 18) ( s181302) d2 y y = 0. ( 19) ( s191317) u = y x 2xyy y 2 + x 2 = 0,., y =. ( 20) ( s201309) (D) (3). f (x) 3f (x) + 2f(x) = 0 (D) f (x) = f(x) f (x) = 2f(x), f(x) (D). (2) f(x) = e x f(x) = e 2x (D). (3) (D) f(x), a, b f(x) = ae x + be 2x. 27

28 ( 20) ( s201326) f(x), f (x) = xf(x),. f (x), f(x). (2) f(0) = 1, f(x). ( 20) ( s201339) = xy x 2, xy. 1,, xy. ( 22) ( s221307) y = 1 N., α, β, N = N(t). dn = αn βn 2 (2), N(t)., N(0) = N 0. ( 22) ( s221311) = 2x + 2y = x + 3y y = 1 x y + xy2., y = x(t). ( 23) ( s231301) ( 24) ( s241301) = 1 x2 x(0) = 0. ( 24) ( s241317) d x(t) = x(t) y(t) + 2z(t) d y(t) = x(t) + 3y(t) d z(t) = x(t) + y(t) + z(t) x(0) = 4, y(0) = 4, z(0) = 1. ( 27) ( s271309) y =, y = d2 y 2, D = d 2xyy = x 2 + y 2 (2) y 7y + 10y = 6x + 8e 2x 28

29 (3) { Dx = 4x y Dy = x + 2y ( 15) ( s151406) , y = d2 y 2, y =. y y 2y = 2x 2 6x (2) x 3 yy = y (3) (y + xy )xy = x , y = d2 y 2, y =. 2x 2 y = x 2 + y 2 (2) y + 2εy + ω 0 2 y = F sin ωx, ε 0, ω 0 2 > ε x(x y) + y2 = 0 (2) ( 16) ( s161405) ( 17) ( s171403) xy = x (3) 2 + y = 2 sin x ( 18) ( s181405) x 2 = x2 + y 2 (2) + y tan x + cot2 x = 0 ( 19) ( s191406) x + 2y = et 3x + 2y = 1 ( 19) ( s191407) (x 2 + 2xy) + (x 2 y 2 ) = 0 (2) 2x y = xy3 (3) 2 2y = 10 cos x (4) x2 d2 y 3x 2 + 4y = x , c. + y = x ( 20) ( s201404) (2) xy = y3 e x 2 (3) e y + xe y = 0 (4) 2 + cy = 0 29

30 ( 21) ( s211403) (a) = tan x tan y (b) cos x y sin x = 2 cos x sin x (c) d2 y 2 + 6y = ex (2) y(t) t, f (t), f (t). (a) y(t). y (t) + k 2 y(t) = 0 (k > 0 ) (b) y(0) = A 0, y (0) = x 2x2 y = y (2) 4y 2 + ( 22) ( s221408) ( ) 2 = 4 (3) d2 y y = 0 (4) d2 y y = sin 3x ( 23) ( s231406) (a) + 2y = x2 (b) d2 y y = 6e x (2) 20, t[ ] T (t)[ ]. (a) k(k > 0), t T (t). (b) t = 0 100, 3 60, x = 3y (2) (3) (4) ( 24) ( s241406) (x 2 y 2 ) 2 = xy ex x + ye x = y = 0 ( 25) ( s251407) x = y 1 (2) x + y = 2y (3) d2 y 2 3 = e3x (4) d2 y 2 2y = sin x ( 26) ( s261404) 30

31 xy = 0 ( ) (2) x + sin x + y = 0 3 = 0 (3) 3 d2 y y = 0 (4) y x ( 27) ( s271407) = y tan x (4) (2) x 2y = x3 e x (3) 2 + y = cos x d 4 y 4 8 d2 y y = x2 ( 28) ( s281407) = e2y x (3) d2 y y = 4ex (2) 2xy = y2 4x 2 (4) sin x 2y cos x = 2x sin3 x ( 29) ( s291406) y + 2y + 10y = e t sin t, y(0) = 1, y (0) = 0 { ( 10 ( s101702) y + 2y + 2y = 0 y(0) = 1, y y(t) (0) = 1 (2) y(t) 0 t 4π ( 11 ( s111705) y = y x log y x ( 12 ( s121702) y + y tan x = 1 cos x, y(0) = 1 ( 13) ( s131704) P (x, y) = 2x m (y + x 2 ), Q(x, y) = x m+1 (1 + x 2 ) + y P y = Q x m (2) m P (x, y) + Q(x, y) = k > 0 = x k+1 (t 0) x(0) = 1 ( 14) ( s141704) x k (t) x k (t) (2) lim k 0 x k (t) ( 15) ( s151702) 31

32 y 2y y + 2y = sin x x = e t, x y, ( 16) ( s161702). x =, x2 d2 y 2 = d2 y 2 (2). x 2 d2 y 2 x 8y = 0 ( 17) ( s171703) (1 + y 2 ) y ( x + (1 y)2 y ) = 0 x 0.175, y y = e x, y(0) = 0, y (0) = 1 ( 18) ( s181703) ( 19) ( s191703) x = x(t), y = y(t), x(0) = 0, y(0) = 1. = 3x + y = x + y = x. (2) = y. (3) x(0) = 0, y(0) = 1. ( 20) ( s201706) = y, = x ( 21) ( s211703) x = 1, y = 1, x = y(1 + y). ( 22) ( s221703) x = 0, y = 1, = xy. (2) x = 0, y = 1, = x + y. ( 24) ( s241706) 32

33 0.180 x = x(t), y = y(t), x(0) = 5, y(0) = 2. = 3x 4y = x 2y ( 25) ( s251704) y = y(x),. = 1 y + y x ( ) u = u(x) u = y. x, ( ) u = u(x). (2) x = 2, y = 4, ( ). ( 26) ( s261703) y = y(x),. y 6y + 11y 6y = 0. (2) y 6y + 11y 6y = e 4x. ( 27) ( s271703) < x <, y(x) y(x) = x 2 +. y(x). x 0 ty(t) ( 28) ( s281706) = y x 3 1 (2) 1 + x3 (3) = (x + y + 1)3 ( 14) ( s141804) ( 14) ( s141805) = x + y x y ( 22) ( s221808) = x 2y 2x + y ( 25) ( s251804) 33

34 x = 0, z = y 2 (0 y 2) z. z, V 0.,. h, V. (2) t t h(t). h(t)., h(t). (3) 45,. ( 29) ( s291801) f(x) = e λx λ a d 4 4 f(x) a4 f(x) = 0 ( 13) ( s132006) C, R, L, q(t), a = R L, b = 1, q(t) CL,. d 2 q 2 + adq + bq = 0 a 2 4b > 0,. ( 19) ( s192002) C,., 90 C 60 C (2x 2y 1) + ( 2x + 6y + 3) = 0 (2) ( 21) ( s212005) 2 6y = 2 cos 2x ( 22) ( s222009) = a sin x (2) y + x = 0, x = π y = a. 2, x = 1 y = 0. (3) x 2 + y = 0, x = 1 y = 1. ( 23) ( s232001) = 1 y , (2). = xy (2) md2 x 2 + kx = 0 (m, k ) ( 26) ( s262008) ( 27) ( s272016) 34

35 y = e5t (a) y = 0 (b) y(t) = Ce αt, C α. (c)., 2. x 2 d2 y 3x 2 + 3y = x5 (2) (d) x = e t, (2). (e) (2) ( ) 4y = 0 ( ) 2 2 4y = xe2x ( ) ( 29) ( s292018) (2) z = (ax 2 + bx)e 2x ( ) a, b (3) ( ) y = 0 2 d2 y y = x ( 3 ( s032105) ( 4 ( s042105) (x + 1)y + xy y = 0 y = ue x u (2) (3) ( 6) ( s062104) = x y = x + 3y : x(0) = 0, y(0) = 1 x = x(t), y = y(t) 35

36 z(t) = x(t) + y(t) z = z(t) (2) z(t) (3) x(t), y(t) ( 7 ( s072103) y y = 0 y y = e 2x cos x ( ) ( ) ( ) (2) y = e 2x (a cos x + b sin x) ( ) a, b (3) ( ) ( 8) ( s082103) y 2y + 2y = d2 y + ay = 0 2 a = 2 (2) a = 2 ( 9) ( s092104) (3) y(0) = y(π) = 0 a ( 11) ( s112103) f(x) u(x, y) = f(x)e 2y 2 u x u y 2 = 0 f(x) y = e 2x sin 3x 2 d2 y 2 ( 12) ( s122103) (2) y d2 y 2 + a + by = 0 a, b ( ) x 2 d2 y 2 + x 4y = x,. ( 13) ( s132105) y = x n ( ) 0 x 2 d2 y 2 + x 4y = 0 n. (2) y = ax ( ) a. (3) ( ). ( 16) ( s162102) : + ay = 0 y(0) = b. a, b. 36

37 (2) f(t), z = f(x + 2y). z z x + z + z = 0, y f(0) = 2 f(t). ( 17) ( s172105) d2 y ay = 0 (a a > 1 ) (2) y(0) = 1, y (0) = 1 (3) y(π) = 0 a ( 18) ( s182104) y (y ) 2 + y = 0 y(0) = 1, y (0) = 0 y = y(x). y. z = log y, z = z(x). (2) y d2 y 2 + ω2 y = 0 ω,.. (2) y(0) = 0, y (0) = 2. (3) y = 0 ω. ( 19) ( s192105) ( 20) ( s202104) {. x (t) = 4y(t) y (t) = x(t). (2) x(0) = 0, y(0) = y = x y. (2) ( 21) ( s212104) ( 22) ( s222104) d2 y 2 + c2 y = 0., c. (2) f(t) 2. 2 z(x, y) = f(3x 4y) 2 z x z + z = 0 f(t). y ( 23) ( s232104) 37

38 u(t) t du u = 0. (2) f(t). 2 z(x, y) = f(x 2 y 3 ) x z x + y z y 5z = 0, f(t) z(x, y) d2 y 2 + 4y = e 2x. (2) 2 z(x, y) = f(x)e 2y 2 z x z y 2 = e 2(x+y), 2 f(x) ( ) = y2 x 2 2xy.. z = y x, z, dz, x. (2) ( ) z x. ( 24) ( s242104) ( 25) ( s252104) (3) (2), ( ) ( ) y + y2 = e x,. ( 26) ( s262103) z = y 2, ( ) z x. (2), ( ) ( ) x 2 d2 y + 4x 2 + 2y = 0. x = e t,., x. (2) 2, x. 2 (3) ( ) x y ( ) y y = e x sin x.. ( 27) ( s272102) ( 28) ( s282103) 38

39 y y = 0. (2) a, b, y = ae x cos x + be x sin x ( ) a, b. (3) ( ) ( 29) ( s292102) (a > 0) (1 + x)y + x(1 y) = 0 (2) = ay, = ax ( 11) ( s112205) t = 0 x = x 0 ( 11) ( s112206) = ex+y (2) + y = 1 ( 29) ( s292209) y 1 (t), y 2 (t) 1 2 = y 2 = y 1 + cos t cos t (2) ( 6) ( s062301) y = f(x) P (x, y) x Q y R P QR y = f(x) (2) P R x y = f(x) (3) (2) (1, 4) y = f(x) x x (2) = + 1) x(y2 y(x 2 + 1) ( 12) ( s122305) ( 13) ( s132304) N 0 α 39

40 N 1 N 1 α t t = 0 (2) N 1 t (3) β N 1 N 2 N 1 + N 2 = N 0 N 1 (4) t N 1 t = 0 N 0 0 (5) ( 15) ( s152304) (2) (3) + y = 1 = 2x + y y + xy2 = 2x 2y + x 2 y (4) (3) y = 3 ( 15) ( s152305) y = 1 (2) 2xy(1 + x) = 1 + y2 (3) = y x + 1 y x ( ) 2 x + y y = 36 11x + y xy + 2y = e x ( 15) ( s152309) ( 16) ( s162305) , 2,. = y + y2 ( 16) ( s162317) (2) (sin x) + (cos x)y = 0 (3) 2 + 2y = 0 ( 17) ( s172306) 40

41 0.232,,, (3), y = y, y, /, / 2 y + 6y + 9y = 0 (2) xy + y 2 = 4 (3) y = 2x 2y + cos x 2x 4y sin y ( 18) ( s182307) , (4) y =., y, y,, 2. y + y 2 = 0 (2) y = 2y + 3y (3) xy (2 + x)y = 0 (4) y(y + 2x) + (y 2 x 2 ) = 0 ( 19) ( s192307) y = f(x)., (3),, (4). x 3 + y = 0 (3) 4x + 2y = 2x + y 1 (2) x = y + x 2 + y 2 (4) y = 0 ( x = 0 y = 0, ) = 7 ( 20) ( s202305) 0.235, (3),, (4) (y + 3x) + (x + 1) = 0 (2) x = 2x ( 1 + x 2) y (3) y y = 0 (4) x e x = 0 (x = 0 y = 1) ,., sinh x = ex e x, cosh x = ex + e x. 2 2 y y = 0 (2) x2 + y 2 2xy = 0 (3) (cosh x) + (y x) sinh x = 0 (4) x2 d2 y 2 + x 4y = x y ( ) 2xy = xy2,. u = y 1 (y 0), ( ) u 1. (2) ( ). ( 21) ( s212305) ( 22) ( s222306) ( 23) ( s232304) 41

42 0.238 y = 1 2 x2. (2) y + y = 2 + 2x. (3) a(> 0), O(0, 0)., P x x H, P x N HN. ( 24) ( s242306) x t x(t), v a v =, a = dv.., x v a. a = 9x, x., x(0) = v(0) = 1. (2) a = 4(v + x), x. (3) a = 4(v + x) + e t, x(0) = 0, v(0) = 3. (4) v = 2tx 2, x(0) = 1. ( 25) ( s252306) , (3), (4). e 2x y x+y + e = 0 (2) (x 3 + y 3 ) 3xy 2 = 0 ( ) 2 (3) (2x + 3y) + 6xy = 0 (4) y = ( x = 0 y = 5, ) = 1 ( 26) ( s262306) x(t). d 2 x 2 = x (2) x(0) = y(0) = 1 x(t), y(t). + = x y = x + y (3) x(0) = y(0) = 1 x(t), y(t). + = x + y = x + y ( 27) ( s272310) ,., y = A sin(nx + α), p(x) q(x)., A, α, n 0. + p(x) 2 + q(x)y = 0 42

43 (2) xy (a, b) R(> 0)., a, b R. (3). 1 x + 2y = 2 ( 29) ( s292306) k α (2) x d f(x) α d f(x) + kf(x) = 0 f(x) 2 f(x) α = y x 3 ( 12) ( s122407) ( 12) ( s122408) x cos y x = x + y cos y x ( 12) ( s122409) m g = 9.8 m/s 2 c = 2 Ns 2 /m 2 ( 13) ( s132410) d 2 x 2 + 4x = 0 (2) d 2 x x = 0 ( 13) ( s132411) xy + (1 y 2 ) = 0 (2) x/ + 2y = x 2 (2) / 2 3/ + 2y = e 3x (4) / 2 + y = cos x (t) u(t) = 1 x(t) = u(t) (x(0) = 0) ( 13) ( s132412) (2) u(t) = 2x(t) + 1 x(t) d 2 x(t) 2 ax(t) = 0 a ( 15) ( s152410) 43

44 x(t) a > 0, a < 0 x(t) x(0) = x 0, (0) = 0 (2) a > 0 lim t x(t) = 0 ( 15) ( s152411) 0.251,. g, t ν,. dν = g k ν ( m, k ) m, ν t., x 2 + y = 0., x 0, y (2) (3) ( 16) ( s162413) ( 16) ( s162414) = ax b (a, b > 0) = x2 a 2 (a > 0) d 2 x 2 + 9x = 5 sin 4t ( 17) ( s172409) m [kg], x, m d2 x 2 = c ( ) 2 mg., g [m/s 2 ], c [kg/m]. lim t ( 17) ( s172410) y = 2xy 2 (2) y 3y + 2y = 3e 2x ( 17) ( s172412) = x t 2 x = 1 (2) x = y2 1 y = 0 (3) x 2 = y2 1 y = 0 ( 18) ( s182407) 44

45 0.257 y = x y (2) y = x 2y ( 18) ( s182408) = y2 1 (2) x(x y) + y2 = 0 (3) x + 2y = x2 (4) d2 y 2 + y = cos x ( 18) ( s182412) k m x t k m d2 x 2 + kx = 0 m x k ω = x = A cos ωt + B sin ωt m (2) m = 0.16(kg), k = 4(kg/sec 2 ) (3) t = 0 x = 2(cm), = 0(cm/sec) A, B 3 ( 18) ( s182418) C 14 C 14 C t 14 C y a = ay 14 C y 0 (2) y 0 6, 000 a (3) 14 C y = 0.125y ,. x = y2 9 ( 18) ( s182420). (2) y = ,. x 2 = y2 + 9 ( 19) ( s192410). (2) y = df(x) = f(x),. ( 19) ( s192411). (2) f(0) = 1. 45

46 ( 19) ( s192413) d 2 x 2 + 9x = 0 (2) d 2 x 2 9x = 0 (3) d 2 x x = 0 ( 20) ( s202407) y + x = 0,. (2) (x 2 2xy y 2 ) + (3y 2 2xy x 2 ) = y(x). 2 d2 y 2 + α + 2y = 4, α,. ( 20) ( s202408) (0) α = 5, y(0) = 0, = 2, y(x). (2) x, α > 0 y(x) y p lim y(x) = y p. y p. x (3) (2) y(x), y(x) α ( 20) ( s202410) x 2 (y 2 1) = 0 (2) cos x = y sin x (3) = y 2 xy x 2 ( 21) ( s212404) 0.268,. + xy = x (x = 1, y = 0) (2) x + x + y = 0 (x = 1, y = 0) (3) = (x + y)2 (x = 1, y = 1) ( 22) ( s222410) xy = y 1 (2) + y = x (3) x y = x 2 + y 2 ( 22) ( s222420) 0.270,. xy = x ( : x = 0, y = 0) 46

47 (2) (3) + ex y = 2e x ( : x = 0, y = 1) + y cos x = sin x cos x ( : x = 0, y = 0) ( 23) ( s232408) f(x) = 5,. (a) y s. + 2a + 5y = f(x) 2 (b) C 1 e αx + C 2 e βx, x 0 a. (2) a = 1, f(x) = 10 sin x, y(0) = 1, = 0,. x=0 α, β ( 23) ( s232410) = ay y t., a., t = 0, y = y 0,. (2) y = 1 2 y 0 t = 5000, a. (3) y = y 0, t = ,. + 2xy = x ( : x = 0, y = 0) (2) y sin x = sin x ( : x = 0, y = 0) (3) x + y = x(1 x2 ) ( : x = 1, y = 0) 0.274, k,, m ( 23) ( s232421) ( 24) ( s242411)., x., t,. m d2 x 2 + kx = 0 k m x k ω =, x = A cos ωt + B sin ωt,. m (2) m = 2.25(kg), k = 4π 2 (kg m/s 2/ m),. (3) t = 0, x = 2(cm), = 0 (cm/s), A, B, 3. 47

48 0.275,. (2) (3) + x2 y = x 2 x = 0 y = 2 = y2 + y x = 0 y = 1 1,, y 2 + y = 1 y 1 y y tan x = sin x x = 0 y = ,. (2) + y = x ( x = 0, y = 1) = y2 1 ( x = 0, y = 0) ( 24) ( s242423) ( 25) ( s252414) ( 26) ( s262411) (2) (3) y = y = ex y = ex. ( 26) ( s262412) e y + (xe y 3y 2 ) = 0 (2) 2 + y = cos x ( 26) ( s262413) = ay (2) d2 x x = cos 2t ( 26) ( s262425) + 2y = x (2) = y + y2 (3) d2 y 2 3 = 0 ( 27) ( s272410) 0.281, ( ). (2) = 2x + y z = y x + 2y x (2x + 2y 1) + x + y + 1 = 0 z = x + y 2 ( 27) ( s272411) 48

49 0.282 y x,,. ( ) y + a = 0 a, y(0) = 1, y (0) = 0. (2) y I = y y 2. I a.,, a I., x, t. t = 0 x 0, x. ( 27) ( s272413) x, α. (2) T 2 α T., ρ ( )., y =. dρ. y = ρ(1 + y 2 ) 1 2 ( 27) ( s272419) 1 (2) x 2 + y 2 + 2x + 2y + 1 = 0, ρ x dρ 0, ρ. (3) ρ, 1., ( y = ν = tan θ π 2 < θ < π ). 2,. ( 27) ( s272425) y sin x = 0 (2) x + 2xy = x (3) (x + y) = y ( 28) ( s282408) = ax y 6 = 5x 2y + 1, (. ), x(t), y(t) x(t) r(t) = R 2., a 5 y(t) 2. t r(t) r 0 R 2, a. 49

50 (2), d(t) = r(t) r 0 =. ( X(t) Y (t) ) R 2, X(t) Y (t) (3) (2) d(t), t + a. ( ) 1 (4) a = 4, r(0) = r(t) x = y + x 2 + y 2 (2) (1 + x) + (1 + y) = , ẋ =. ( ẋ = 1 x ) x 1 K, K. ( 0 0 ) ( 28) ( s282410) ( 28) ( s282413), t, x. (2) t = 0, x = K. t - x. 10 x O t (3) y + 1 x y = ex, y = 3 (x > 0) = x2 + 1 x. (2) (2x + y) + (x + 2y) = 0. ( 28) ( s282424) ( 17) ( s172504) (3) x y = x log x. ( 17) ( s172506) x, y,. ( 17) ( s172507) (3) = ky + cos ωx (2) + 1 y2 1 x 2 = 0 = y x + x2 y 3 u = y 2. ( 18) ( s182508) 50

51 0.293 (x, y) A A y x 2 + y 2 = cx (c ) ( 18) ( s182509) = (x + y + 2)2 ( 18) ( s182511) xy = 0. (2) + 2xy = x + x3, y(0) = 1. ( 19) ( s192506) = 2y (2) = y(1 2y) (3) = 2y + sin x (4) d2 y y = 0 (2) ( 19) ( s192508) d 2 x 2 + ω 0 2 x = 0., ω 0. d 2 x 2 + ω 0 2 x = cos(ωt)., ω,, : (a) ω ω 0 ; (b) ω = ω = xex+y. (2) + cos x sin x y = 1 cos 2 x y ( π ( 19) ( s192509) ) = 2. ( 20) ( s202503) = y(1 x) (2) = y(1 y) (3) = y(1 y2 ) ( 20) ( s202505) (E 1 ) x 2 d2 y 2 2y = x, (E 2 ) x 2 d2 y 2 2y = 0, (E 1 ).. (E 2 ), x = e t. (2)., (E 2 ). (3) (2), (E 1 ) x 2 + y2 = 0, (2) y x = x3, (x 0) ( 20) ( s202506) ( 21) ( s212506) 51

52 y = e R { p(x) + P (x)y = Q(x) R } P Q(x)e (x) + C., C = x 2y 4 2x + 4y (2) = x 2y 4 2x 4y ( 21) ( s212507) (3) d5 y 5 d4 y 4 d3 y d2 y 2 2y = 0 ( 21) ( s212508) y = y(x). + y y2 = 0 y(0) = 1 (2) + y = x y(0) = 0 2 ( 22) ( s222507) = 2xy (2) = 2xy(1 y) (3) = x + y x y ( 22) ( s222508) (y + 2xy) + x = 0 1 e 2x 1. (2) y = 0 2. (2) y + y = u 3, u 4. du + 3u = 0 4 (3) 4. 2 ( 22) ( s222509) (4) 3 (2) u y 1, 3. ( 22) ( s222510) y = y(x). = y2 sin 2x cos 3x (x 0) y(0) = 5 (2) 3 = x + 2y x y = 1 (x 1) ( 23) ( s232507) 52

53 = y sin x (2) = y(1 y)(2 y) (3) = y + sin x ( 23) ( s232508) , y =, y = d2 y 2. xy + 2y + xy = 0 y 1 = cos x, 1. x (2) x u, y = uy 1 1, u 1. (3) du = v v. (4) u. (5) 1. d 2 u 2(tan x)du 2 = , y =, y = d2 y 2. y + 4y + 3y = 0, y(0) = 0, y (0) = 1. (2) y + 4y + 4y = 0, y(0) = 0, y (0) = 1. (3) y + 4y + 5y = 0, y(0) = 0, y (0) = y = y(x) ( ). 2 + y = x2, d2 y 2 + y = 0 (2) y(0) = 0, ( 23) ( s232509) ( 23) ( s232510) (0) = 0 ( 24) ( s242503) x = 2y (2) sin x + y cos x = x (3) (2x2 y + y 2 ) + (x 3 + xy) = g(x, y) h(x, y) m g(x, y) = h(x, y) ( 24) ( s242508)., m, t f(x, y). f(tx, ty) = t m f(x, y) ( 24) ( s242509) 53

54 x 2 d2 y 2 + x y = 0 x 2 d2 y 2 + x y = x 1. (2) y = x log x 2. 2 (3) y = y(x)., log. ( 24) ( s242510) x y = x log x (x > 0) (2) d2 y y = e2x ( 25) ( s252506) ( x 2 y 2) y = xy ( 28) ( s282502) y = cos t t = 0 y = e 2π + e π t = π y = 1.9 ( 10) ( s102601) y = 0 (2) y(0) = 1 2, (3) a, b (0) = 2 y 1 (x), y 2 (x) 2 + a + by = 0 f(x) = 1 y 2 2 y 1 f(x) x 0 f(x 0 ) 0 x f(x) y + y 2y = 0 (2) y(0) = 4, y (0) = 1 ( 12) ( s122601) ( 13) ( s132607) 54

55 y + 4x = 0 (2) x + y = sin x ( 13) ( s132608) y(0) = 1, y (0) = 1 y + 3y + 2y = ( 15) ( s152605) 3 = 0 (2) xy 3y = 5 ( 16) ( s162603) , y x, y, y. y y 2 = 0 (2) y + 3y + 2y = y 4y 3y = 0 (2) y = 4xy x ( 16) ( s162604) ( 17) ( s172601) , x = 0 y = y 0. = a by (a, b ) ( 17) ( s172608) x 1 x + xex = 0 ( 18) ( s182603) x = 1 y = 1 = y2 x d2 y y = 25e2x = (1 y)y,. y(0) = a., a. ( 18) ( s182612) ( 18) ( s182620) (2) y(x), lim x y(x). ( 19) ( s192605) 0.331,. x 2 + y = 0, x = 1 y = 1. ( 19) ( s192617) 55

56 0.332 y = e2x y(0) = 1. ( 19) ( s192624) y = y 2 + 2y 3 (2) y + 6y + 10y = , (2) (7). ( 20) ( s202603) d2 x 2 = a. y =, y t + b (2). (3)., =, y x (4). (5).,, ν[m/s] 37.5 ν + 50 [m/s2 ]., ν = 0 [m/s] 0.75[m/s 2 ],. 75[m/s](= 270[km/h]) (6) [s], (7) [m] y x,. ( 20) ( s202611) y +y = 0 (2) y 7y +12y = 6x 2 +5x+18 (3) y 4y +4y = cos x y x. ( 20) ( s202616),. + y cos x = sin x cos x y(0) = 0. (2) y(x) 0 x π. ( 21) ( s212606) 0.337, x, α α x. t f(t)., f(0) = β. ( 21) ( s212613) x 2 y + 2y = 0 (2) y 6y + 8y = 0 ( 21) ( s212616) y(x), (A),. y (x) tan(x) y(x) = 2e 2 sin(x), π/2 < x < π/2 (A) 56

57 y(x; y 0 ) y(0) = y 0 (A), y(x; y 0 )., y 0. (2) y(x; y 0 ), lim y(x; y 0) y 0. x π/2, y 0 lim y(x; y 0).,, x π/ y(x) ( 22) ( s222605) y + 2y + ay = 0., a. ( ) a = 1, ( ). (2) a > 0, ( ) y lim y(x) = 0. x y = y(x), y = (x). (E) y + y cos x = sin x cos x.. y + y cos x = 0. (2) (E). ( 23) ( s232607) (3) (E), y(0) = 0. y(x) (4) (3) y, lim x 0 x 2. ( 28) ( s282604) (3) y + P (x)y = Q(x) P (x), Q(x) x u, v x y = uv v = e P (x) dv + P (x)v = 0 du/ F (x) du = F (x) u = F (x) + C (C ) y 57

58 dv + P (x)v = 0 (2) du/ F (x) P (x), Q(x) (3) x + y = log x y log ( 11 ( s112705) y(x) i = y = 0 ( 12) ( s122705) + y = x (2) y = e2x ( 18) ( s182703) R,., h.., h R. A h. R (2) V h. (3). V, dv/.,, h A. (4) (3),, Sk h S, k.,, h t., t = 0 h = R, T. ( 25) ( s252703) (x) (x) + 2y(x) = 0 (2) (x) 2 + y(x) = cos x ( 17) ( s172804) x( 1 2xy)y = 2y(1 + xy) (2) y + 2y + 5y = exp x y ( 18) ( s182802) y 5y + 6y = 0, y. ( 20) ( s202803) 58

59 0.349., = y. y + 2y + y = e t. (2) y(0) = 0, y (0) = y + 6y + 5y = 5x. (2) y + x 1 x y 1 x y = xe x ( 23) ( s232801) (i) 0 1 y 1 = e x. (ii) y = ue x., u x. ( 27) ( s272802) , k, p, q, p q,, t = 0 x = 0. = k(p x)(q x) ( 29) ( s292804) y +2y 3y = e x x y = e x z z +az +bz = x a, b (2) z + az + bz = x ( 9 ( s092902) y + y = 0 (2) w = w(x) 4xw + 2w + w = 0 ( ) x x = t 2 (3) ( ) t u = w(t 2 ) u ( 11) ( s112905) x = e t x 2 d2 y + px + qy = R(x) 2 x t + (p 1) 2 + qy = R(et ) (2) x 2 d2 y 3x + 4y = x2 2 ( 12) ( s122904) 59

60 0.355 x y y = 1 + y (2) y x 1 (t y(t)) ( 13) ( s132905) x > 0 ( )x 2 y + xy y = 0 y 1 = x ( ) (2) y = y 1 z y ( ) z (3) y 1 ( ) ( 15) ( s152902) x 0. x 2 y = (y 2 + 1)(y 1)(y + 2) y = x (2x + y)y (x + 2y) = 0 ( 16) ( s162904) ( 17) ( s172903) (2) y(0) = f(x) [0, ),., f(x). f (x) 1 x + 1 f(x) = (x + 1)2 e x (2) f(0) = 1 f(x) x = 0, y = 1, = 1 + x + y y = c 0 + c 1 x + c 2 x c n x n ( 18) ( s182908) ( 21) ( s212907) ( 22) ( s222908) y + αy + βy = γe x, (α 0, β 0, γ 0 ) y = e 2x + (1 + x)e x.,. 60

61 α, β, γ. (2). ( 23) ( s232909) P (x, y) + Q(x, y) = 0, P (x, y) = Q(x, y), y x. f(x) (0, ) f(π) = 1. ( ) y sin x f(x) + f(x) = 0, x > 0 x,. f(x). (2). ( 24) ( s242908) y = e x xy + p(x)y = x., y =.. p(x). (2). (3) : x = ln 2, y(x) = 0., ( 25) ( s252909) y + 3y + 2y = 3 y, y y = y(x) y(0) = 1, y (0) = 2 12 ( s123002) y = f(x) x 2 = y a2 y = a 2 (b + 1) (a > 0, b > 0),.. (2) x = 0 y = 0, + y = x ( 14) ( s143113) ( 15) ( s153107) = 0, x = π y = b, y a y = y(x). ( 16) ( s163109) y = 5 ( 16) ( s163110) ( 1 + x 2) = xy. ( 17) ( s173102) 61

62 0.371., e, a, b. f(x) = 0. (2) f(x) = e bx. + ay = f(x) ( 17) ( s173109) = y x (2) y = 0 ( 19) ( s193104) (2). (2) xf(x) = = y y = f(x)., f(0) = 1. x 1 1 f(x) + 1 y = f(x). x ( 21) ( s213101) (3),. (2) (3) = 3y = 2 y x x = 0 y = 5 x = 1 y = 3 = cos(x + y) x = 0 y = 0 ( 21) ( s213105) y(x) + xy = x, y(0) = 0. (2) y(x) d2 y y = (3x 2 y y 3 ) = (3y 2 x x 3 ) ( 21) ( s213111) (2) P (x, y) + Q(x, y) = 0 { P (x, y) + Q(x, y) } P (x, y) = c y c, (3x 2 y y 3 ) = (3y 2 x x 3 ). ( 22) ( s223101) 62

63 0.377., x 1. (x 1) + y 1 = 0 ( 22) ( s223107) d2 y 2 + y = cos x, y(0) = 0, y (0) = 0, y (x) (x)/. ( 23) ( s233107) ẍ + γẋ + kx = 0,., ẍ = d2 x 2, ẋ =., x(0) = 1, ẋ(0) = 0. γ = 0, k = 9. (2) γ = 2, k = 9. (3) γ = 6, k = 9. ( 23) ( s233110) (t) = t + x(t) t, x = 3 ( 23) ( s233114) m., k, g., m, k, g. v(t).,. (2) v(0) = 0,., v(t),. ( 24) ( s243106) y 2y + 5y = x, y(0) = 1, y (0) = 0., y =, y = d2 y 2. ( 25) ( s253104) y x 1 y, y(0) = 1. y + 3y = cos 2x ( 25) ( s253106) y = 0 ( 26) ( s263105) , y =, y = d2 y 2. y + 4y + 3y = e 2x. ( 28) ( s283105) 63

64 0.386 y = y(x) 1 x y = x. ( 29) ( s293106) = + ay y = f(x) (3)., a a > 0. f(0) = a,, f(x) x. (2) g(x) = xf(x), y = g(x). (3). lim a 0 0 g(x) f(x) ( 29) ( s293113) d2 y y = cos 2x. ( 29) ( s293117) (a) di(t) + λi(t) = v 0 (a) t 0 λ v 0 0 v 0 = 0 C I(t) = C exp( λt) (2) v 0 0 C (a) I(0) = 0 I(t) t ( 13) ( s133208) t = 0 x = x 0 v 0 k d 2 x 2 k2 x = 0 (2) x f(x) d 2 x 2 + k2 x = 0 d 2 f(x) 2 ( 14) ( s143206) + df(x) 2f(x) = ex ( ) f(x) g(x) ( ) 0 h(x) d 2 h(x) 2 + dh(x) 2h(x) = 0 f(x) = g(x) + h(x) g(x) = Axe x (A : ) A 64

65 (2) h(x) y(ρ) (ρ) dρ ρ { (ρ) dρ + 1 ( 15) ( s153205) } l(l + 1) ρ 2 y(ρ) = 0., ρ, 0 ρ <, l., y(ρ) y(ρ) = u(ρ) ρ,, u(ρ) d 2 u(ρ) dρ 2. p(ρ). + p(ρ)u(ρ) = 0 (a) (2), (a) ρ ρ 0 u(ρ). (a) ρ p(ρ), u(ρ)., u(ρ), λ u(ρ) = e λρ (b). λ, u(ρ). ρ 0 p(ρ), u(ρ)., u(ρ), λ u(ρ) = ρ λ. λ, u(ρ) ( 16) ( s163205)., a. (2). 2 + a2 y = Ae ibx, A, a, b, i. (a) y = Be ibx. (b). 2 + a2 y = t x(t). ( 19) ( s193208) d 2 x. 2 + ω 0 2 x = 0, ω 0. d 2 x (2) λ + ω 0 2 x = 0, ω 0 λ. λ 2 ω 2 0 = ω 2 < 0 ω :. (3) (2). ( 20) ( s203207) 65

66 d 2 x 2 + ω2 x = 0 x(t)., C 1, C 2. (2) x(t)., B 1, B 2 C 1, C 2. x(t) = C 1 sin(ωt) + C 2 cos(ωt) x(t) = B 1 e +iωt + B 2 e iωt ( 22) ( s223205) d 2 x 2 + ω 0 2 x = A cos(ωt) ( ) x(t), A = 0 x 0 (t) A 0 x 1 (t) x 0 (t) + x 1 (t).., A, ω, ω 0. x 0 (t). (2) x 1 (t) = α cos(ωt) + β sin(ωt) ( ), α β x 1 (t)., ω ω 0. (3) x(0) = x 0, = 0, ( )., ω ω 0, t=0. ( 24) ( s243203) = y + 2 x + 1 (2) + y = sin x (3) d2 y 2 2y = 0 ( 25) ( s253206) = (y + 1)(x2 + 2x) (2) x = ae iωt, m d2 x 2 = kx λ., a ω. (a), x. (b) d2 x, x. 2 (c) ω. (d), ω. 66

67 ( 26) ( s263203) d 2 x 2 + µ + ω2 x = 0, µ, ω.. x(t) exp( at). x(t) a, µ ω. (2) a = a R ± ia I,. µ ω., a R a I, i = 1. (3) (2), x(0) = 0, = 1, a R t=0 a I (2) d2 x 2 ( 27) ( s273203) = x + sin t, t = 0 x = 0 x(t). = 1 ( ) 2. (a) v =. t = 0 v = 0 v(t). (b) t = 0 x = 0 v = 0 x(t) m d2 x 1 2 = k 1x 1 + k 2 (x 2 x 1 ) m d2 x 2 2 = k 1x 2 k 2 (x 2 x 1 ), m, k 1, k 2.. ( 28) ( s283207) X 1 = x 1 + x 2, X 2 = x 1 x 2 X 1, X 2 k1., ω = m, k1 + 2k 2 ω =, X 1, X 2. m (2) t = 0, x 1 = 1, x 2 = 0 v 1 = 0, v 2 = 0 x 1 (t), x 2 (t)., v 1 = 1, v 2 = 2. ( 29) ( s293206) ( ) 2 = d2 y 2 + (2) x = 0 y = 0, = 0 (or 0.5) ( 10) ( s103303) = x a y(t) (x t) ( 11) ( s113302) 67

68 ( ) ( ) d2 y d d + 9y = 7 cos 3x 2 + 3i 3i y = 7 cos 3x ( ) ( ) d d z = 3i y + 3i z = 7 cos 3x dz + 3iz = 7 cos 3x z (2) z 3iy = z y ( 14) ( s143302) P (x) Q(x) x. + P (x)y = Q(x). y = e [ P ] P Qe + c (2) (a) 2x + y = 2x2 (b) (1 + x 2 ) = xy + 1 ( 16) ( s163302) 0.406,,,,.,. (3). t y(t), t = 0,, y 0. (a) y(t), y (t) + λy(t) = 0 λ. (b) y(0) = y 0, (a). (c), t = 0 1 n. n (2), t = 0, y 0,, y 0 T. (a),, t = mt m. (b) m,, y s. y s. (3), t = 0, Y,, T y d. 2 Y, 2 y d. ( 25) ( s253301) 68

69 0.407, ( ). 2y = x2 + 2x (x = 1 y = 0) (2) = ay + b y (x = 0 y = 1), a, b a 0 (3) = (1 y2 ) tan x (x = 0 y = 2) ( 26) ( s263301) t > 0,, ( ) x = x(t). = 1 (t = 0 x = 1) 1 + x 0.409, ( ). ( 26) ( s263302) x + y = 0 (x = 0 y = 2) (2) = 2x + y y (x = 1 y = 1) ( 27) ( s273301) V,. t C(t), Q,, t δt t + δt., C(t) C(t + δt).,,. C(t + δt) =. δt 0, C(t). C(t) =., t = 0, C (3). ( 27) ( s273302) t P (t),,. dp = ap, a. 0 P 0 (> 0), P (0) = P 0, 1 P. 1 69

70 (2) 1,. P max, (1 P/P max ). ( dp = a 1 P ) P P max P., a P 0. (3) (2). 2 ( 28) ( s283301) x y = 0, y2 = Cx C. x = 1 y = , ( ). = y2 + y x = 0 y = 3 7x 3y + 2 (2) = 3x 4y + 5 x = 0 y = 1 (3) (2x y + 1) + (2y x 1) = 0 x = 0 y = y + y = x y y = e x, ( 29) ( s293301) ( 29) ( s293302) ( 11) ( s113404) ( 12) ( s123408) y y 2y = 0 y(0) = 2, y (0) = xyy = x 2 + y x 2 y = x 2 + xy + y 2 ( 13) ( s133407) ( 14) ( s143407) ( 15) ( s153408) y x x = e t y t = x, 2 = x2 d2 y 2 + x (2) x 2 d2 y 2 2y = 0 ( 16) ( s163410) y + 2y + 5y = 10 sin x. a cos x + b sin x a, b. (2) y(0) = 1, y (0) = 0. ( 17) ( s173410) 70

71 0.421 y 2y 3y = 3x 2 + x x (2) d2 y y = 0. ( 18) ( s183405) (2) d2 y y = 1, y(0) = 1, y (0) = ( ) = y2 x 2 2 y = u, ( ) u. x (2) y = 3 ( ). ( 19) ( s193405) ( 21) ( s213404) y = y(y 1) x. ( 23) ( s233404) ( ). ( ) (2) y(0) = 1 ( ). = 2x(y2 + 1) y = y(x) { yy + (y ) 2 + yy = x y(0) = 1, y (0) = 0 ( 24) ( s243404). z = z(x) z = yy. (e x z) x. (2) z x. (3) y x. ( 26) ( s263404) y + 4y + 4y = 0 y = y(x), y(0) = 0, x 2. ( 27) ( s273404) a. 2 y + ay = 0, 2 y (0) = 0, y = 0. ( 28) ( s283404) 71

72 F (x), (F (x) + F (x)) sin x = F (x) sin x F (x) cos x + C., C. (2) n 3. y + y = e x { x n + n(n 1)x n 2} sin x y = y(x) y(0) = 0. ( 29) ( s293405) x y = y(x) y 2y + 5y = 0 (2) y 2y + 5y = 4e x (3) y 2y + 5y = 4xe x (4) y 2y + 5y = 4e x + 4xe x y(0) = 0, y (0) = 1 ( 9 ( s093503) a 2 + by = 0 y = y(x) b = 2a 2, (2) b = a2, (3) b = 2a2 4 a ( 13) ( s133502) a(t), b(t) I x 1 (t) 0 I x (t) + a(t)x (t) + b(t)x(t) = 0 x 2 (t) I x 2 (t) = x 1 (t) t 1 t 0 {x 1 (τ)} 2 exp( (2) x 1 (t), x 2 (t) (0, ) x(t) x(t) = t x(t) a, b > 0, a b u (t) + a u(t) = 0 (0 < t < ) v (t) + b v(t) a u(t) = 0 (0 < t < ) u(0) = 1 v(0) = 0 0 τ t 0 sin(2(t u)) x(u)du + t a(s)ds)dτ. ( 13) ( s133503) ( 13) ( s133504) 72

73 u(t), v(t) (2) v(t) (3) v(t) ( 14) ( s143503) 0.435, log x, e. y(x) : y + 2y + y = 0. (2) z(x) (x > 0) x 2 z + 3xz + z = 0 ( ). x = e t, t = log x z(e t ) = w(t). (3) ( ). (4) ( ) :. z = 0, e 1 z(x) = 1 ( 17) ( s173505) α β, y (x) + αy (x) + βy(x) = 0 ( ).,. ( ). (2) φ(x) x 0 φ(x 0 ) = φ (x 0 ) = 0 ( )., φ(x) = 0. (3) (2), x 0, C 1, C 2 φ(x 0 ) = C 1 φ (x 0 ) = C 2 ( ). ( 17) ( s173508) x 2 y xy + y = f(x) (A) x > 0 f(x) = 0 y 1 = x (A) (2) u y = uy 1 w w = u (A) w x (3) f(x) = 0 (A) (4) f(x) = x 2 x (A) ( 18) ( s183503) 73

74 0.438 y(x) xy (x + 1)y + y = 2x 2 e 2x (A) (A) xy (x + 1)y + y = 0 y = e px (p ) (2) y = e px u (p u x ) (A) u (3) (2) (A) ( 18) ( s183505) x (t) + a(t)x (t) + b(t)x(t) = 0. ( ) a(t), b(t) x 1, x 2 ( ) J [ x 1 (t) J(t) := det x 1 (t) x 2 (t) x 2 (t) ] J : J (t) = a(t)j(t). (2) x 1, x 2 ( ) [ x 1 (0) x 2 (0) J(0) = det x 1 (0) x 2 (0) ] 0 t (x 1 (t), x 1 (t)), (x 2 (t), x 2 (t)) (3) x 1, x 2, x 3 ( ) t x 1 (t) x 2 (t) x 3 (t) det x 1 (t) x 2 (t) x 3 (t) = 0 x 1 (t) x 2 (t) x 3 (t) (c 1, c 2, c 3 ) (0, 0, 0) t c 1 x 1 (t) + c 2 x 2 (t) + c 3 x 3 (t) = y = 2y + y 2.. ( 18) ( s183508) (2) y = 3,.. ( 19) ( s193502) 74

75 0.441 x (t) + ax(t) = sin t ( )., a. x (t) + ax(t) = 0. (2) ( ). (3) a > 0. ( ) x(t) x(0) = α, x = β., α, β. ( 19) ( s193509) = y + a(x 3 + xy 2 ) = x + a(x 2 y + y 3 ), t = 0 (0, 0). a. (x(t), y(t)). a = 0, t. (2) a > 0, t > 0. (3) a < 0, t > 0, t (0, 0). ( 20) ( s203504) C P (x, y)., P C y. P C x Q. C, P Q Q x,. C. ( 21) ( s213503) y (x) + y(x)(4e 2x 1) = 0 (x R) ( ). t = e x z(t) = y(log t). (2) z(t) = c k t k+ρ (ρ R, c 0 0). k=0 ρ c k (k = 1, 2, )., ρ = ±1/2. (3) c 1 = 0. (2) c k, z 1 (t), z 2 (t). (4) ( ) y 1 (x), y 2 (x) e x (y 1 (x) 2 + y 2 (x) 2 ) = 1 ( x R) d2 y y = 0. ( 21) ( s213505) 75

76 (2) d2 y y = cos ωt y = P cos ωt + Q sin ωt, P Q., ω ω > x = x(t). = 2x2 + t 2. t > 0 ( 22) ( s223503) v(t) = {x(t) t 1 } 1 v(t)., dv t v. (2), v(t). (3) C(t) v(t) C(t). (4) x(t). (5) x = x = x(t), y = y(t) = 2xy = x 2 y 2, t = 0 (x(0), y(0)) = (a, b)., a 0. ( 22) ( s223505) (x(t), y(t)) C, x 2 + y 2 = Cx. (2) t x(t), a, b ,. (t) = 2x(t) 3y(t) + F (t) (t) = y(t) 2x(t) F (t) = 0, x(t), y(t). ( 22) ( s223508) (2) F (t) = e 2t, y(t) y 1 (t) = Ae 2t., A. (3) F (t) = e 2t, x(t), y(t). (4) F (t) = e 2t, x(0) = 2, y(0) = 0 x(t), y(t) q 1 = q 1 (t), q 2 = q 2 (t), q 3 = q 3 (t) ( 23) ( s233502) q 1 = q 2 + q 3 2q 1 q 2 = q 3 + q 1 2q 2 q 3 = q 1 + q 2 2q 3 76

77 ,, f q 1 (0) = f, q 1 (0) = 0 q 2 (0) = f, q 2 (0) = 0 q 3 (0) = f, q 3 (0) = 0 d2 y y = 0 y = y(x). (2) d2 y y = x2 + 3x + 1 y = y(x). (3) d2 y y = x2 + 3x + 1 y = y(x) x = 0, y = 10 = ( 23) ( s233508) ( 24) ( s243502) x 2 d2 f(x) 2 + ax df(x) + bf(x) = 0, (x > 0). t = log x g(t) = f(e t ). a, b, log. (2). (3). x 2 d2 f(x) 2 x 2 d2 f(x) x df(x) + 3x df(x) 3f(x) = 0, (x > 0) 3f(x) = log x, (x > 0) ( 24) ( s243508) x 2 d2 y 2x 2 + 2y = 0 x = et, y(x)., x > 0. (2) x 2 d2 y 2x 2 + 2y = 6x4 y = Ax 4, A. (3) (2) y(x)., x = 1, y = 4 = x = x(t), y = y(t), = (x2 + y 2 )y + kx = (x2 + y 2 )x + ky, t = 0 (x(0), y(0)) = (x 0, y 0 )., x y 0 2 = 1. ( 25) ( s253502) 77

78 X(t) = x 2 (t) + y 2 (t). X(t),. (2) x(t), y(t) t, k, x 0, y 0. ( 25) ( s253505) m, k 2. x 0, x(t) { x (t) + x(t) t m x(t) k = 0 (t > 0), x(0) = x 0.. y(t) = x(t) 1 k, y(t). (2). 0 t m e (k 1)t (3) ( ) x(t), lim t t 0 0 x(t) = t 0 x(t). x(t) x 0. ( 26) ( s263504) ( ) C : y = u(x) (x, y) y 2xy 2 1,, C 1, 3, u(x). ( ) ( 27) ( s273502) a, b, c b 2 ac > 0, b > 0. D = b 2 ac.. + bx + cy = 0, t 0 ax by = 0, t 0 x = x(t), y = y(t) x(0) = 0, y(0) = 1 b, c, D t. (2) y = y(t), y(t) > 0 t 0. (3). z(t) = x(t) y(t), t 0 dz + az2 + 2bz + c = 0, z(0) = 0 dz(t) (4) lim z(t) lim b, c, D. t t x(t), y(t),. = 2x 2y + cos 2t = x ( 28) ( s283504) 78

79 x(t) y(t). (2) x(0) = y(0) = 0 x(t) y(t). (3) t x(t) A cos(ωt + θ), A, ω, θ., A, ω, θ, A > 0, ω > 0, 0 θ < 2π., θ ,. = y2 2 y x 1 2x 2 (x 0) ( ) u(x) = xy, u(x). (2) ( ) y = y(x). ( 28) ( s283509) (3) ( ) y = y(x), x = 1 y = 2. ( 29) ( s293502) y = y(x) y (x) (x 2 1)y(x) = 0, y(0) = 1, y (0) = 0 z(x) = y (x) + xy(x) z z(0) (2) z(x) (3) (2) y(x) 13 ( s133603) y = cos(x) (2) 2 + y = cos(x) xy + 4 3y + 4x = y y 2 (2) ( 15) ( s153601) d 3 y d2 y 2 2y = e2x ( 17) ( s173602) ( y ) = f x (2) = x2 + y 2 2xy ( 18) ( s183602) = ( a x + y ) 2 (a > 0) (2) d2 y y = 6e2x ( 19) ( s193602)

80 = x + 3y x (2) y = 1 2 x2 + x + e x ( 20) ( s203602) a d2 x 2 + b + cx = 0, a > 0, c > = 4x3 y ( 22) ( s223607) ( 22) ( s223612) (x + 1) xy = 0 (2) d2 y y = ex ( 23) ( s233608) xy = x (2) d2 y 2 + = 0 ( 25) ( s253601) (2) = y2 y y = sin x ( 25) ( s253607) (1 + x 2 ) + (1 + y 2 ) = 0 (2) d2 y y = e3x ( 28) ( s283602) = x + y (2) d2 y y = e2x ( 29) ( s293602) = 3x y x + y u = y x (2) (3) = x (6x 2 y 2 2) y (x 2 + y 2 + 2) u y 4y + 3y = 9x ( 6) ( s063802) ( 11) ( s113805) 80

81 0.474 d 2 f df df (x) 3 (x) + 2f(x) = 0, f(0) = 2 (0) = 1 13 ( s133807) t x(t) x + x 2 + a(t)x + b(t) = 0 x = x(t) = u (t) u(t) u(t) (2) x = x(1 x) ( 14) ( s143803) y + 4y + 4y = x 3 ( 15) ( s153807) x(t), y(t) x (t) = y(t), y (t) = x(t) x(0) = a, y(0) = b ( 16) ( s163806) a, b. y y (a + b)y + aby = 0. (2) y = y 2.. ( 17) ( s173803) y y = 0 (2) y + y = 0 ( 19) ( s193806) y = y(x) y + (5 x 2 )y = 0.,. y = ze x2 /2, z = z(x). (2) z, z. ( 19) ( s193809) y = emx (m R) (2) y = 0 ( 20) ( s203809) , y, y, 2. y y 2y = 0 (2) y y 2y = cos x ( 21) ( s213808) 81

82 0.483 f (x) + f(x) = sin x, f(0) = f (0) = 0, F (x) = f(x) cos x f (x) sin x, G(x) = f(x) sin x + f (x) cos x.. F (x), G (x). f. (2) F (x), G(x). (3) f(x). ( 21) ( s213812) 0.484,. y = y2 ( ) z = y 1, z. (2). (3) x = 0, y = 1/2, ( ). ( 22) ( s223805) y y = e x y(0) = a, y (0) = b., x 0 a b. y =, y = d2 y 2. ( 23) ( s233805) x = x(t) t C.,. d 2 x 2 + ( ) 2 4 = 0, x(0) = (0) = 0. ( 24) ( s243805) y 2y = xe 2x., y =, y = d2 y 2. y = (Ax 2 + Bx)e 2x A, B, A, B. (2) y = y(x),., y =, y = d2 y 2. 2y 5y + 2y = 0, (2) 2y 5y + 2y = e x, ( 26) ( s263809) ( 27) ( s273806) 82

83 0.489., x. y(x) u(x) 1{ u(x)y(x) } = y (x) + x 2 y(x) u(x). (2) y(x) y (x) + x 2 y(x) = x 5. (3) p(x), q(x), y(x). y (x) + p(x)y(x) = q(x) ( 1, 1) C - f(x) n, a n = f (n) (0) n! (1 x 2 )f (x) xf(x) = 1, f(0) = 0.. ( 29) ( s293805) a n. 1 (2) (1 x 2 ) n 1 π., 1 x e x (0 x 1) 0 2 n π e x2 = 0 2. (3) n {a n }.,. ( 30) ( s303805) (1 x 2 ) d2 y 2x 2 + 6y = 0 ( 13) ( s133906) y = 0 ( 16) ( s163904) y x,. y = xy, y(0) = 2. (2) y 2y + 2y = 0, y(0) = 1, y (0) = m t v(t), m dv(t) = mg cv(t) (m, g, c ) ( 17) ( s173907) v(t),, t = 0 v = 0 ( 18) ( s183905) 83

84 (3x + y + 3) + (x + 3y + 2) = 0 (2) y 4y + 3y = 2x ( 19) ( s193912) xy = 0 (2) y = sin 2x ( 20) ( s203907) x 2 d2 y 2x 2 + 2y = 0. ( 21) ( s213901) d2 y + y = cos x. 2 ( 21) ( s213902) = x(1 x) x(0) = x 0 (> 0) x 2 y + xy2 + x = 0 (2) ( 21) ( s213903) y = 4x + 6e3x ( 21) ( s213909) = 1 y2 (2) d2 y 2 2y = cos x ( 23) ( s233908) x y,. (x + 3)y + xy 2 = 0 (2) y 6y + 8y = x ( 24) ( s243901) (x 2 + 2) xy = 0 (2) y = 4e x ( 25) ( s253903) c 1, c 2 y (1 x 2 )y 2xy = 0, y(0) = c 1, y (0) = c 2 n y (n) (0) (2) y y 1 y 2. ( 18) ( s184003) 84

85 (2) p dp = y 2y3 (0 y < 1), p(0) = 0 p C 1 ( [0, 1) ). { y y + 2y 3 = 0, 0 y < 1, x R, (3) (2). lim y(x) = 1, x a, b a 0 e ax sin bx, e ax cos bx (2) 1 + ay = cos bx ( ) ( 19) ( s194003) (3) y(0) x ( ) y(x) a b , lim t u(t). (3) du + u = 1, u(0) = 0 (2) du t 2 u = 0, u(0) = 1 du = u(1 u), u(0) = (2) d 2 x 2 + 4x = 0, d 2 x + 4x = sin 3t, 2 ( t = 0, x = 1 ) = 0 ( t = 0, x = 0 ) = 0 ( 18) ( s184105) ( 20) ( s204102) ( 20) ( s204107) u(r) (0, ) 2,, u (r) (0, ). f(x, y).. f(x, y) = u(r) (, r = x 2 + y 2 ) 2 f x 2 (x, y) + 2 f y 2 (x, y) = u (r) + 1 r u (r). (2) 2 f x 2 (x, y) + 2 f (x, y) = 0, y2 u(r) = a log r + b (a, b ) , x = 1 y = 2. x + y = 2x ( 24) ( s244103) ( 24) ( s244112) 85

86 0.511 d2 y y + 3 = 0. ( 25) ( s254102) y = 0 ( 26) ( s264105) ay = b., a, b 0. ( 24) ( s244202) 0.514,., a, b 0.. = a bx (2) t = 0 x = 0. (3) t, (2). ( 25) ( s254203) y + 2y + 3y = 2 cos x ( 11) ( s114303) = log x x = 1 y = 1 ( 12) ( s124302) xy + y = x 2 ( 13) ( s134312) = 2xy ( 13) ( s134313) y = xu x(x y) + y2 = 0 ( 13) ( s134314) % I 0 (2) = y x (2) = y x u = y x ( 14) ( s144302) ( 15) ( s154307) 86