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1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 + x ) = 1 + y (11) x(y ) = y 1

1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (7) 3x + y + (x + y) (8) = x + y 1 x + y = 0

1.3 (1)-(9), Bernoulli (10)-(1) (1) + y = x () x + y = x (3) + y x = 1 x (4) y = sin x (5) + y tan x = sec x (6) + y cos x = cos x sin x (7) x + y = x ln x (8) (1 + x ) + xy = x (9) (1 x ) + xy = 1 (10) x = xy + y (11) x + y + (x 1)y = 0 (1) = xy + x3 y 3 3

1.4 (1) (y + x) + x = 0 () (6x y + 1) + (y x 3) = 0 (3) x(x + y) + (x y ) = 0 (4) xy + (y x ) = 0 (5) y(x + y) + x(x + 3y) = 0 (6) (y xy) x = 0 (7) (xy cos x) + (x 1) = 0 (8) (tan y 3x ) + x sec y = 0 (9) (tan y x) + sec y = 0 (10) (x + ln y) + x y = 0 1.5 p = y (1) y = px p () y = px p + p (3) y = xp + a p (4) y = xp + p (5) y = x p + p (6) yp = x + p 3 (7) p 3 4xyp + 8y = 0 (8) p 3 + axp + x 3 = 0 (9) (a x )p 3 + bx(a x )p p bx = 0 (10) p + py cot x y = 0 4

1.6 (1) y + (x 1)(y ) 3 = 0 () xy + x(y ) y = 0 (3) (1 + x )y + 1 + (y ) = 0 (4) (xy y ) = (y ) + 1 (5) y y = e x (6) y = e y ( ) (7) yy + (y ) = 1 (8) yy (y ) + (y ) 3 = 0 (9) y (y ) 3 = 1 (10) y(y 1)y + (y ) = 0 (11) y y = 3(y ) (1) 5(y ) = 3y y.1 1. t N(t) λ(> 0) N(t). N(0) = N 0 3. λ T log = 0.6931 4. t = 1 N(t) N 0 99% log 100 = 4.605, log 99 = 4.595 5

. A, B A 1 B 1 3 C 1 A B (t = 0) A a B b C 0 b > a t C t.3 10 [ C] t T (t)[ C] dt dt = k(t 10) (k > 0; ) (1) t = 0 60 [ C] 10 55 [ C] 40 [ C] log = 0.6931, log 3 = 1.099, log 5 = 1.609, log 10 =.303.4 M OX T MT OX OT.5 (r, θ).6 E E = E 0 sin ωt () ω = πf E i R L i 6

e (R/L)t L sin ωtdt = R + L ω e(r/l)t sin(ωt φ), tan φ = Lω R (3).7 Clairaut.8 ( ) ax + by + c = f αx + βy + γ (4) a, b, c, α, β, γ.9 1. v m(v) m u v v. v = 0 M 0 dm dv = m V (V : ) (5) 7

3 3.1 (1) y 6y + 8y = 0 () y 10y + 5y = 0 (3) y + 8y + 5y = 0 (4) y y 6y = 6x 1 (5) y y + y = x (6) y y + y = 6x 6 (7) y y + y = x 4 + x 3 1 (8) y 6y + 9y = e 4x (9) y y 8y = e x (10) y + y 6y = e 3x (11) y 3y + y = cos x (1) y + y = cos x cos 3x (13) y y + y = e x sin x (14) y y + y = (x + 1)e 3x (15) y + y = sec x 3. (1) y y y + y = 0 () y + y y y = 0 (3) y 3y + 3y y = e x (4) y + y + y = 0 8

4 Q T (x) k d T + Q = 0 (6) k Q (3.49) (3.5) 1. (5) T (x) = C 3 x (C 3 ) C 3 (5). (5) 3. x = 0, h T 1 T (0) = T (h) = T 1 T (x) 9

5 (1) y = (y + y)e x, x = 0 y = 0 () y + (cos x) y = sin x cos x, x = 0 y = 0 (3) (x + ln y) + x = 0, x = 1 y = 1 y 6 (1) y x x 1 y + 1 x 1 y = x 1, y 1 = x, () y + y + y = e x y = e x 7 (1) () (3) (4) = x y, dt dt = 5x y, dt = 7x + y, dt dt = 5x y + et, = 4x 3y dt = x 3y dt = x 5y dt = x 3y + et 10

8 1.1 (1) (1 + x)(1 + y) = C () y = Ce 1 x (3) y = C x + 1 (4) xy(y + 3) = C (5) ax a y = C(a + y ax a ) (6) x 1 + y 1 = C, y = 1 (7) sec x + tan y = C (8) sin y cos x = C (9) y + 1 = +C(y 1) cos x (10) y x = C(1 + xy) (11) x y = Ce y 1.(1) x + xy + y = C () Cx = (x 4y ) 5 (3) y (x + y ) = Cx 6 ( y (4) x sin = C x) (5) x Cy = C ( ) y + 1 (6) tan 1 = ln (x ) x + (y + 1) + C (7) 3x + xy + y = 4x + C (8) y x 1 ln x + y 3 = C 11

1.3 (1) y = Ce x + x 1 () y = 1 3 x + C x (3) xy = C + x x4 4 (4) y = Ce x 1 (sin x + cos x) (5) y = sin x + C cos x (6) y = sin x 1 + Ce sin x (7) 4xy = C x + x ln x (8) C y = 1 + 1 + x (9) y = x + C 1 x (10) x y = C ln x, y = 0 (11) 1/y = x + C x + 1 (1) y (x 1 + Ce x ) + 1 = 0 1.4 (1) x + xy = C () 3x xy + y + x 3y = C (3) x 3 + 3x y y 3 = C (4) x + y = Cy (5) x y + xy 3 = C (6) y + x = Cx y (7) x y sin x y = C (8) x tan y x 3 = C (9) tan y = Ce x + x 1 (10) 1 3 x3 + x ln y = C 1

1.5 (1) y = Cx C, y = 1 4 x () y = Cx C + C, y = 1 8 x + 1 4 x 1 8 (3) y = Cx + a C, y = ± ax (4) x = 3 p + C p, y = 3 p + C p, p [ ( (5) x = ln p + ) ] p p + 1 + C p + 1, [ ( y = ln p + ) ] p 1 + 1 + C p + 1 + p, p (6) x = p + Cp p 1, y = + C p 1 + p, p (7) y = C(C x), y(4x 3 7y) = 0 (8) x = at 1 + t 3, y = a (1 + 4t 3 ) 6(1 + t 3 + C, p = tx, ) p (9) (y + bx + C)[x a sin(y + C)] = 0 (10) (y sin x) Cy + C = 0 13

1.6 (1) y = ln (x 1) ± (x 1) + C 1 + C () y = ln x + C 1 + C, y = C (3) y = 1 x + 1 (C 1 + 1 ) ln C 1 C 1 C 1 x + 1 + C, C 1 y = 1 x + C (4) y = C 1 C 6 x3 ± 1 + 1 x + C x + C 3, [ 13 (1 x ) 3/ + x sin 1 x + ] 1 x y = ± 1 + C 4 x + C 5 (5) y = C 1 e x + C x + C 3 x + C 4 e x (6) C1e y = cosh(c 1 x + C ) 1 C3e y = cos(c 3 x + C 4 ) + 1, y = ln x + C (7) y = x + C 1 x + C (8) x = y C 1 ln y + C, y = C (9) y = 1 (x C ) C 1 + (x C ) + C 1 ln (10) y ln y = C 1 x + C, y = C 1 (11) y = + C 3 C 1 x + C x C C1 + (1) y = C 1 x + C + C 3 x + C 4, y = C 5 x + C 6 x + C 7 (x C ) + 1 C 1 + C 3 3.1 (1) y = C 1 e 4x + C e x () y = (C 1 + C x)e 5x (3) y = e 4x (C 1 cos 3x + C sin 3x) (4) y = C 1 e 3x + C e x + x (5) y = (C 1 + C x)e x + x + (6) y = e x (C 1 cos x + C sin x) + 3x (7) y = e x (C 1 + C x) + x 4 + 10x 3 + 48x + 13x + 167 (8) y = (C 1 + C x)e 3x + e 4x 14

(9) y = C 1 e 4x + C e x 1 8 ex (10) y = C 1 e x + C e 3x 1 5 xe 3x (11) y = C 1 e x + C e x 3 10 sin x + 1 10 cos x (1) y = C 1 cos x + C sin x + 1 x sin x + 1 cos 3x 8 (13) y = e [(C x 1 1 ] x) cos x + C sin x (14) y = e x (C 1 + C x) + 1 8 e3x (x 4x + 5) (15) y = C 1 cos x + C sin x + x sin x + (cos x) ln cos x 3. (1) y = C 1 e x + C e x + C 3 e x () y = C 1 e x + C e x + C 3 e x (3) y = C 1 e x + C xe x + C 3 x e x + e x 3 (4) y = e x/ (C 1 cos x + C sin 3 x) + e x/ (C 3 cos 3 3 x + C 4 sin x) 5.(1) y = 0 () y = sin x 1 + e sin x (3) 1 3 x3 + x ln y = 1 3 6.(1) y = C 1 x + C e x (x + x + 1) () y = C 1 e x + C xe x + 1 x e x 7.(1) x = C 1 e t + C e t, y = C 1 e t + 4C e t () x = (C 1 + C t)e 4t, y = (C 1 + C + C t)e 4t (3) x = e 6t (C 1 cos t + C sin t), y = e 6t [(C 1 + C ) cos t + (C C 1 ) sin t] (4) x = (C 1 + C t)e 4t + 4 5 et 1 36 et, y = (C 1 + C + C t)e 4t + 1 5 et + 7 36 et 15

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