n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)

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1 D d dx n d n y a 0 dx n + a d n 1 y 1 dx n a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y (n 1) a n 1 y + a n y = 0...(3) (3) 1 (2) y(x) (3) y 1 (x) (2) 1 y 2 (x) y(x) = y 1 (x) + y 2 (x) (i)(3) y 1 (x) (ii)(2) 1 y 2 (x) 2 (3) (3) y 1 (x) n Y 1 (x),..., Y n (x) C 1,..., C n y 1 (x) = C 1 Y 1 (x) C n Y n (x) 1

2 n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1) 1.2 (3) y y ay = 0...(4) (4) dy y = dy dx = ay dy y = ax adx log y = ax + c y = e ax+c = Ce ax e ax a (i) e ax 2

3 (ii) a = α + βi e iθ = cos θ + i sin θ e ax = e (α+βi)x = e αx (cos βx + i sin βx) n (3) y = e ρx ρ a 0 ρ n + a 1 ρ n a n 1 ρ + a n = 0...(5) (5) (3) ρ 1,..., ρ n (ρ i ρ j, i j) ρ i (i = 1,..., n) (i)ρ i y = e ρ ix (ii) ρ j = α j ± β j i y ± = e (α j±β j i)x = e α j e ±β jix = e α jx (cos β j x ± i sin β j x) y = e αjx cos β j x, e αjx sin β j x 1.y 2ay + a 2 y = 0 ( ρ 2 2aρ + a 2 = (ρ a) 2 = 0 ρ = a 1 y = e ax y = u(x)e ax y = u (x)e ax + au(x)e ax, y = u (x)e ax + 2au (x)e ax + a 2 u(x)e ax u (x)e ax +2au (x)e ax +a 2 u(x)e ax 2a(u (x)e ax +au(x)e ax )+a 2 u(x)e ax = 0, 3

4 , u (x)e ax = 0 u (x) = 0 u(x) = Ax + B e ax, xe ax (4) dy dx ay = 0 D = d dx ( d dx a)y = 0 (D a)y = 0 (2) (3) (a 0 D n + a 1 D n a n 1 D + a n )y = f(x)...(2) (a 0 D n + a 1 D n a n 1 D + a n )y = 0...(3) (3) (5) 2.y y 2y = 0 (D 2 D 2)y = (D 2)(D + 1)y = (D + 1)(D 2)y (D+1)y = 0 (D 2)y = 0 (D 2 D 2)y = 0 (D + 1)y = 0 (D 2)y = 0 (4) 4

5 e x, e 2x (D 2 D 2)y = 0 2 e x, e 2x (D a)(e ax y) = e ax Dy 3.(D a) n y = 0. ( (D a) n y = (D a) n (e ax e ax y) = (D a) n 1 (D a)(e ax e ax y) = (D a) n 1 e ax D(e ax y) = (D a) n 2 e ax D 2 (e ax y) =... = e ax D n (e ax y) e ax D n (e ax y) = 0 D n (e ax y) = 0 n e ax y = (n 1) y = x n 1 e ax, x n 2 e ax,..., xe ax, e ax n 1 3 n = 2 3 (3) (i) ρ i m i m i. {x m i 1 e ρ ix, x m i 2 e ρ ix,..., xe ρ ix, e ρ ix }, 5

6 (ii) ρ j = α j ± iβ j m j {x m j 1 e α jx cos β j x, x m j 2 e α jx cos β j x,..., xe α jx cos β j x, e α jx cos β j x}, {x m j 1 e α jx sin β j x, x m j 2 e α jx sin β j x,..., xe α jx sin β j x, e α jx sin β j x} 2m j 1.3 (2) (2) f(x)=e kx 4. (D a)y = e kx (i) (i)k a y = 1 k a exk (ii)k = a y = xe xk (D a)y = (D a)(e ax e ax y) = e ax D(e ax y) (D a)y = e kx e ax D(e ax y) = e kx 6

7 D(e ax y) = e (k a)x k a e ax y = e (k a)x dx = 1 k a e(k a)x 1 (ii) y = 1 k a ekx (D a)y = e ax e ax D(e ax y) = e ax D(e ax y) = 1 e ax y = dx = x y = xe kx 5. (a 0 D n + a 1 D n a n 1 D + a n )y = e kx 7

8 F (D) = a 0 D n + a 1 D n a n 1 D + a n F (k) 0 y = 1 F (k) ekx ( D a 0 D n + a 1 D n a n 1 D + a n = a 0 (D α 1 )(D α 2 )...(D α n ) α i (a 0 D n + a 1 D n a n 1 D + a n )y = e kx a 0 (D α 1 )(D α 2 )...(D α n )y = e kx 1 α 1 k (D α 2 )...(D α n )y = 1 a 0 (k α 1 ) ekx α 1 k, α 2 k,...α n k y = (D α n )y = 6. 1 a 0 (k α 1 )(k α 2 )...(k α n 1 ) ekx 1 a 0 (k α 1 )(k α 2 )...(k α n 1 )(k α n ) ekx = 1 F (k) ekx (D k) n y = e kx y = xn n! ekx (D k) n y = e kx 8

9 (D k) n 1 (D k)y = e kx (D k) n 1 e kx D(e kx y) = e kx (D k)e kx D n 1 (e kx y) = e kx e kx e kx D n (e kx y) = e kx n D n (e kx y) = 1 e kx y = xn n! y = xn n! ekx y = u(x)e kx u(x) 2 (a 0 D n + a 1 D n a n 1 D + a n )y = e kx F (D) = a 0 D n + a 1 D n a n 1 D + a n F (D) = (D k) m G(D), G(k) 0 y = 0! = 1 xm G(k)m! ekx 9

10 7. (1)y + y 2y = e 2x (2)y y 2y = e 2x (3)y 4y + 4y = e 2x ( )F (D) = D 2 + D 2, F (2) = = 4 0 y = 1 4 e2x (2)F (D) = D 2 D 2 = (D 2)(D + 1) y = x (2+1) e2x = x 3 e2x (3)F (D) = D 2 4D + 4 = (D 2) 2 y = x 2! e2x = x 2 e2x ( (a 0 D n + a 1 D n a n 1 D + a n )y = e kx F (ρ) = a 0 ρ n + a 1 ρ n a n 1 ρ + a n F (k) 0 y = Ae kx F (ρ) = (ρ k) m G(ρ), G(k) 0 y = Ax m e kx y = Ae kx y = Ax m e kx A F (2) = = 4 0 y = Ae 2x y = Ae 2x y = 2Ae 2x, y = 4Ae 2x 4Ae kx = e 2x A = 1 4 (2)F (ρ) = ρ 2 ρ 2 = (ρ 2)(ρ + 1) y = Axe 2x y = Axe 2x y = Ae 2x + 2Axe 2x, y = 4Ae 2x + 4Axe 2x 4Ae 2x + 4Axe 2x Ae 2x + 2Axe 2x 2Axe 2x = e 2x, 10

11 3Ae 2x = e 2x A = 1 3 (3)F (ρ) = ρ 2 4ρ + 4 = (ρ 2) 2 y = Ax 2 e 2x y = Ax 2 e 2x y = 2Axe 2x + 2Ax 2 e 2x, y = 2Ae 2x + 8Axe 2x + 4Ax 2 e 2x 2Ae 2x + 8Axe 2x + 4Ax 2 e 2x 4(2Axe 2x + 2Ax 2 e 2x ) + 4Ax 2 e 2x = e 2x, 2Ae 2x = e 2x, A = f(x) = sin ωx( sin ωx e iθ = cos θ + i sin θ (a 0 D n + a 1 D n a n 1 D + a n )y = cos ωx( sin ωx)...(1) (a 0 D n + a 1 D n a n 1 D + a n )Y = e iωx...(2) Y y 1, y 2 (a 0 D n + a 1 D n a n 1 D + a n )(y 1 + iy 2 ) = cos ωx + i sin ωx...(3) y 1, y

12 (a 0 D n + a 1 D n a n 1 D + a n )y 1 = cos ωx...(4) (a 0 D n + a 1 D n a n 1 D + a n )y 2 = sin ωx...(5) (2) (a 0 D n + a 1 D n a n 1 D + a n )y = e kx ; k 3 (a 0 D n + a 1 D n a n 1 D + a n )Y = e iωx F (D) = a 0 D n + a 1 D n a n 1 D + a n F (D) = (D iω) m G(D), G(iω) 0 Y = x m G(iω)m! eiωx (4), (5) Y = e iωx = cos ωx + i sin ωx xm G(iω)m! eiωx x m (cos ωx + i sin ωx) G(iω)m! 8. y 3y + 2y = cos x( sin x) y + 4 = cos 2x( sin 2x) ( F (D) = D 2 3D + 2 F (i) = i 2 3i + 2 = 1 3i 0 y 3y + 2y = e ix 12

13 Y = 1 1 3i eix = ( i)(cos x+i sin x) = 1 10 cos x 3 10 sin x+i( 1 10 sin x+ 3 cos x) 10 y 1 = 1 10 cos x 3 sin x, 10 y 2 = 1 10 sin x cos x y 3y + 2y = cos x, y 3y + 2y = sin x F (D) = D F (2i) = 0 F (D) = D = D + 2i)(D 2i) y + 4y = e i2x Y = x 4i ei2x = 1 4 ix(cos 2x + i sin 2x) = 1 4 x sin 2x 1 ix cos 2x 4 y 1 = 1 x sin 2x, 4 y 2 = 1 x cos 2x 4 y + 4 = cos 2x, y + 4 = sin 2x 13

14 iω F (D) = 0 F (D) D 2 + ω 2 F (D) = D 2 + ω 2 k G(D) G(iω) 0 ( (a 0 D n + a 1 D n a n 1 D + a n )y = cos ωx( sin ωx) F (ρ) = a 0 ρ n + a 1 ρ n a n 1 ρ + a n F (iω) 0 y = A cos ωx + B sin ωx F (ρ) = (ρ iω) m G(ρ), G(iω) 0 y = x m (A cos ωx + B sin ωx) y = A cos ωx+b sin ωx y = x m (A cos ωx+ B sin ωx) A, B F (D) = D 2 3D + 2 F (i) = i 2 3i + 2 = 1 3i 0 y = A cos x + B sin x y = A sin x + B cos x, y = A cos x B sin x A cos x B sin x 3( A sin x+b cos x)+2(a cos x+b sin x) = cos x( sin x), (A 3B) cos x + (B + 3A) sin x = cos x( sin x), 14

15 { A 3B = 1 B + 3A = 0, { A 3B = 0 B + 3A = 1 A = 1 10, B = 3 10, B = 1 10, A = 3 10 y = 1 10 cos x 3 10 sin x, y = 1 10 cos x sin x F (ρ) = ρ = ρ + 2i)(ρ 2i) y = Ax cos 2x + Bx sin 2x y = 2Ax sin 2x + A cos 2x + 2Bx cos 2x + B sin 2x, y = 4A sin 2x 4Ax cos 2x + 4B cos 2x 4Bx sin 2x 4A sin 2x 4Ax cos 2x + 4B cos 2x 4Bx sin x + 4 Ax cos 2x + Bx sin 2x = cos 2x( sin 2x), 4A sin 2x + 4B cos 2x = cos 2x( sin 2x), { A = 0 4B = 1, { 4A = 1 4B = 0 A = 0, B = 1 4, B = 0, A = 1 4 y = 1 4 x sin 2x, y = 1 x cos 2x 4 15

16 1.3.3 f(x)=e kx cos ωx( e kx sin ωx) 2 (a 0 D n + a 1 D n a n 1 D + a n )y = e kx cos ωx( e kx sin ωx) (a 0 D n + a 1 D n a n 1 D + a n )Y = e (k+iω)x Y Y = y 1 + iy 2 y 1, y 2 (a 0 D n + a 1 D n a n 1 D + a n )y 1 = e kx cos ωx (a 0 D n + a 1 D n a n 1 D + a n )y 2 = e kx sin ωx 4 (a 0 D n + a 1 D n a n 1 D + a n )Y = e (k+iω)x F (D) = a 0 D n + a 1 D n a n 1 D + a n F (D) = (D (k + iω)) m G(D), G(k + iω) 0 Y = x m G(k + iω)m! e(k+iω)x y 1, y 2 e iωx = cos ωx + i sin ωx xm G(k+iω)m! e(k+iω)x xm G(k+iω)m! ekx (cos ωx + i sin ωx) 9.y 3y + 2y = e 3x cos 2x( sin 2x) 16

17 F (D) = D 2 3D + 2 F (3 + 2i) = (3 + 2i) 2 3(3 + 2i) + 2 = 2 + 6i 0 y 3Y + 2Y = e (3+2i)x Y = i e(3+2i)x Y = ( i)e3x (cos 2x + i sin 2x) = 1 20 e3x cos 2x + 3 ( 20 e3x sin 2x + i 3 20 e3x cos 2x 1 ) 20 e3x sin 2x y 1 = 1 20 e3x cos 2x e3x sin 2x = 1 20 e3x (cos 2x 3 sin 2x) y 2 = 3 20 e3x cos 2x 1 20 e3x sin 2x = 1 20 e3x (3 cos 2x + sin 2x) ( (a 0 D n + a 1 D n a n 1 D + a n )y = e kx cos ωx(e kx sin ωx) F (ρ) = a 0 ρ n + a 1 ρ n a n 1 ρ + a n F (k + iω) 0 y = e kx (A cos ωx + B sin ωx) F (ρ) = (ρ (k + iω)) m G(ρ), G(k + iω) 0 y = x m e kx (A cos ωx + B sin ωx) 17

18 y = e kx (A cos ωx + B sin ωx) y = x m e kx (A cos ωx + B sin ωx) A, B F (D) = D 2 3D + 2 F (3 + 2i) = (3 + 2i) 2 3(3 + 2i) + 2 = 2 + 6i 0 y = e 3x (A cos 2x + B sin 2x) y = e 3x (A cos 2x + B sin 2x) y = 3e 3x A cos 2x + 3e 3x B sin 2x 2e 3x A sin 2x + 2e 3x B cos 2x, y = 5e 3x A cos 2x + 5e 3x B sin 2x 12e 3x A sin 2x + 12e 3x B cos 2x 5e 3x A cos 2x + 5e 3x B sin 2x 12e 3x A sin 2x + 12e 3x B cos 2x 3(3e 3x A cos 2x + 3e 3x B sin 2x 2e 3x A sin 2x + 2e 3x B cos 2x) + 2e 3x (A cos 2x + B sin 2x) = e 3x cos 2x( sin 2x),, 2e 3x (A 3B) cos 2x 2e 3x (B + 3A) sin 2x = e 3x cos 2x( sin 2x), { A 3B = 1 2 B + 3A = 0, { A 3B = 0 B + 3A = 1 2 A = 1 20, B = 3 20 B = 1 20, A = 3 20 y = e 3x ( 1 20 cos 2x sin 2x) y = e3x ( 3 20 cos 2x + 1 sin 2x) 20 18

19 1.3.4 f(x) 10. y y 2y = 48x 2 76x + 44 y y = 48x 2 76x + 44 ( (D 2 D 2)y = 48x 2 76x + 44 y = (D 2 D 2) 1 ( 48x 2 76x + 44) ( 48x 2 76x + 44) ( 2 D + D 2 ) ( 48x 2 76x + 44) ( 2 D + D 2 ) = 24x x 5 (D 2 D)y = 48x 2 76x + 44 (D 1)Dy = 48x 2 76x + 44 Dy = Y Y (D 1)Y = 48x 2 76x + 44 Y = (D 1) 1 ( 48x 2 76x + 44) = ( 48x 2 76x + 44) ( 1 + D) = 48x x Dy = 48x x y = 16x x x 19

20 ( (a 0 D n + a 1 D n a n 1 D + a n )y = (x n ) F (ρ) = a 0 ρ n + a 1 ρ n a n 1 ρ + a n (i)f (0) 0 y = x n (ii)f (0) = 0(0 m ) y = x (n + m) 11.y y 2y = x 2 + x + 1 ( y = Ax 2 + Bx + C y = 2Ax + B, y = 2A, 2A (2Ax + B) 2(Ax 2 + Bx + C) = x 2 + x + 1, 2Ax 2 (2A + 2B)x + (2A B 2C) = x 2 + x + 1, 2A = 1 2A 2B = 1 2A B 2C = 1 A = 1, B = 0, C = 1 2 y = 1 2 x2 1 20

21 1.3.5 f(x)=e kx (x 1 (i)k a (D a)y = xe kx y = 1 k a (xekx 1 k a ekx ) (ii)k = a y = 1 2 x2 e ax (D a)(e ax e ax y) = e ax D(e ax y) e ax D(e ax y) = xe kx D(e ax y) = xe (k a)x (i)k a e ax y = xe (k a)x dx = 1 k a {xe(k a)x e (k a)x dx} = 1 k a (xe(k a)x 1 k a e(k a)x )...( ) y = 1 k a (xekx 1 k a ekx ) (ii)k = a ( ) e ax y = xdx = 1 2 x2...( ) y = 1 2 x2 e ax (D a)y = x n e kx 21

22 ( )( ) e ax y = x n e (k a)x dx...( ) e ax y = x n dx = xn+1 n ( ) ( ) x n e (k a)x dx I n = x n e (k a)x dx n I n = x n e (k a)x dx = xn k a e(k a)x n x n 1 e (k a)x dx k a = xn k a e(k a)x n k a I n 1 I n ( ) k a k = a ( ) y = xn+1 n+1 eax (D a) 2 y = xe kx (k a) (D a) 2 y = xe kx (D a)(d a)y = xe kx (D a)y = Y Y (D a)y = xe kx (k a) Y Y = 1 k a (xekx 1 k a ekx ) y (D a)y = 1 k a (xekx 1 k a ekx ) { (D a)y = 1 k a xekx...(1) (D a)y = 1 (k a) e kx...(2) 2 (1) y 1 (ii) y 1 = 1 k a { 1 k a (xekx 22

23 1 k a ekx 1 )} = (k a) xe kx 1 2 (k a) e kx (2) y (i) y 2 = (k a) e kx y 3 y = y 1 y 2 = 1 (k a) xe kx 1 2 (k a) e kx 1 3 (k a) e kx = 1 3 (k a) xe kx 2 2 (k a) e kx 3 Y (D a) 2 y = xe ax (D a)y = Y (D a)y = xe ax (ii) Y = 1 2 x2 e ax y (D a)y = 1 2 x2 e ax y = x2+1 e ax = 1 6 x3 e ax y 4y = xe 2x ((D 2 4)y = (D + 2)(D 2)y (D + 2)(D 2)y = xe 2x (i) (D 2)y = 1 4 (xe2x 1 4 e2x ) 2 (D 2)y 1 = 1 4 xe2x (ii) y 1 = x2 e 2x 23

24 (D 2)y 2 = 1 16 e2x (ii) y 2 = 1 16 xe2x y = y 1 y 2 = 1 8 x2 e 2x 1 16 xe2x e kx (x n (a 0 D n + a 1 D n a n 1 D + a n )y = e kx (x n ) F (ρ) = a 0 ρ n + a 1 ρ n a n 1 ρ + a n (i)f (k) 0 y = e kx (x n ), (ii)f (ρ) = (ρ k) m G(ρ), (G(k) 0, k m ) y = e kx (x n + m ) cos ωx( sin ωx) (x n ) 2 y + P (x)y + Q(x)y = R(x)...(1) 24

25 R(x) = 0 y + P (x)y + Q(x)y = 0...(2) 2.1 y + P (x)y + Q(x)y = 0 (1) y 1 (x), y 2 (x) y 1 (x), y 2 (x) (1) y(x) c 1, c 2 y(x) = c 1 y 1 (x) + c 2 y 2 (x) 5 y 1 (x) y 1 (x) e R P (x)dx y 2 (x) = y 1 (x) y 2 (x) dx y + ay + a2 4 y = 0 1 y = e ρx ρ ρ 2 + aρ + a2 4 = 0 (ρ + a 2 )2 = 0 ρ = a 2 1 y = e a 2 x R y 2 (x) = e a e 2 x P (x)dx y 2 (x) dx = e a 2 x e R adx e ax dx = e a e 2 x ax e ax dx = xe a 2 x 25

26 2.2 y + P (x)y + Q(x)y = R(x) y + P (x)y + Q(x)y = R(x) y(x) y + P (x)y + Q(x)y = 0...(1) Y 1 (x) = c 1 y 1 (x) + c 2 y 2 (x) y + P (x)y + Q(x)y = R(x)...(2) 1 ()y 0 (x) y(x) = c 1 y 1 (x) + c 2 y 2 (x) + y 0 (x) (2) 6 (2) y 0 (x) (1) y 1 (x), y 2 (x), y 0 (x) = y 1 (x) R(x)y2 (x) W (y 1, y 2 )(x) dx + y 2(x) R(x)y 1 (x) W (y 1, y 2 )(x) dx W (y 1, y 2 )(x) y 1 (x), y 2 (x) x 2 y 3xy + 3y = 2x 3 x 2...(1) (1) x 2 y 3xy + 3y = 0...(2) y = x k y = kx k 1, y = k(k 1)x k 2 (2) k(k 1)x k 3kx k + 3x k = 0 k(k 1) 3k + 3 = 0 k 2 4k + 3 = 0 (k 3)(k 1) = 0 k = 1, 3 26

27 (2) y 1 = x, y 2 = x 3 W (x, x 3 x x 3 ) = 1 3x 2 = 2x3 y 1 R(x) = 2x 1 R(x)y2 (x) y 0 (x) = y 1 (x) W (y 1, y 2 )(x) dx + y 2(x) R(x)y 1 (x) W (y 1, y 2 )(x) dx (2x 1)x 3 y 0 (x) = x 2x 3 dx + x 3 = x( 1 2 x x) + x3 ( 1 + log x) 2x = 1 2 x2 ( x (log x) x) (1) (2x 1)x 2x 3 dx y = C 1 x + C 2 x x2 ( x (log x) x) = C 1 x + C 2 x 3 + x 2 (1 + (log x) x) y 1 = x y = xu(x) y = xu (x) + u(x), y = xu (x) + 2u (x) x 2 y 3xy + 3y = 2x 3 x 2 x 2 (xu (x) + 2u (x)) 3x(xu (x) + u(x)) + 3xu(x) = 2x 3 x 2 x 3 u (x) x 2 u (x) = 2x 3 x 2 u 1 x u = 2 1 x u =U U 1 x U = 2 1 x = dx x U 1 x U = 0 du U U = x U = c(x)x c (x)x + c(x) c(x) = 2 1 x c (x) = 2 x 1 x 2 U = x(2 log x + 1 x ) + Cx. u = (2x log x + 1) + Cx u = 2( x2 2 y = 2( x2 2 log x + x2 4 )x + x2 + Cx 3 + Dx c(x) = 2 log x + 1 x. log x + x2 4 ) + x + Cx2 27

28 y = x k p.147 x x = e t x 2 d2 y dx 3x dy 2 dx + 4y = x2 x = e t y(t) ( dy dx = dy dt dt dx = dy dt 1 dx dt = dy 1 dt e = t dy dt e t, x d2 y dx = d 2 dx ( dy dx ) = d dt ( dy dx ) dt dx = d dt ( dy dt e t )e t = ( d2 y dt 2 e t dy dt e t )e t = ( d2 y dt 2 dy dt )e 2t x 2 d2 y dx 2 3x dy dx + 4y = x 2 e 2t ( d2 y dt 2 dy dt )e 2t 3e t dy dt e t +4y = e 2t d2 y dt 4 dy 2 dt y = t2 +4y = e2t 2! e2t = t 2 2 e2t d2 y dt 4 dy 2 dt + 4y = 0 e2t, te 2t y = t2 2 e2t + C 1 e 2t + C 2 te 2t. x y = x2 2 (log x)2 + C 1 x 2 + C 2 x 2 log x (1) 1 y 1 (x) (1) y 2 (x) (2) (1) 1 y 1 (x) 7 y 1 (x) (1) (2) y 0 (x) y 0 (x) = y 1 (x) φ(x)dx R P (x)dx φ(x) = e y 2 (x) y 1 (x)r(x)e R P (x)dx dx y 1 (x) (1) e R P (x)dx y 2 (x) = y 1 (x) y 2 (x) dx y 1 (x) (1) (2) y(x) e R P (x)dx y(x) = c 1 y 1 (x) + c 2 y 1 (x) y 2 (x) dx + y 1 (x) R P (x)dx φ(x) = e y 2 (x) y 1 (x)r(x)e R P (x)dx dx φ(x)dx, (x + 1)y + xy y = 1...(1) 28

29 (x + 1)y + xy y = 0...(2) y 1 y 1 = e kx y 1 y = y 1 u(x) (2) y 1 = e kx y 1 = ke kx, y 1 = k 2 e kx (2) k k 2 (x + 1)e kx + kxe kx e kx = 0 k 2 (x + 1) + kx 1 = 0 (k 2 + k)x + (k 2 1) = 0 k 2 + k = 0 k 2 1 = 0 k = 1 y = e x u(x) y 1 = e x y = e x u + e x u y = e x u 2e x u + e x u (x + 1)y + xy y = 1 (e x u 2e x u + e x u )(x + 1) + ( e x u + e x u )x e x u = 1 u = U e x u (x + 1) e x (x + 2)u = 1 u (x + 1) (x + 2)u = e x U (x + 1) (x + 2)U = e x du dx (x + 2)U x + 1 = ex x + 1 U (x) = e x + C 1 e x (x + 1) 29

30 u(x) = U (x) dx = ( e x + C 1 e x (x + 1))dx = e x + C 1 e x x + C 2 y = e x u(x) = e x ( e x + C 1 e x x + C 2 ) = 1 + C 1 x + C 2 e x y = e kx. (1)x 2 y 3xy + 4y = 2x 3 + x 2 (2)x 2 y 2xy + 2y = 2x + 2 (3)(1 + x 2 )y 2xy + 2y = 1 x2 x (4)x 2 y (x + 2)xy + (x + 2)y = x 4 e x (5) x 2 y + xy + y = x(hint; (1) (5)y = x k ) (6)xy (2x + 1)y + (x + 1)y = (x 2 + x 1)e 2x (7)(x + 1)y (3x + 4)y + 3y = (3x + 2)e 3x (8)xy (2x 1)y + (x 1)y = xe x (hint; (4) (6)y = e kx ) 2.3 y = y 2,x = 0 y = c y = a n x n = a 0 + a 1 x + a 2 x a k x k +... n=0 y = na n x n 1 = a 1 + 2a 2 x ka k x k n=1 y 2 = a 2 0+(a 0 a 1 +a 1 a 0 )x+...+(a 0 a k +a 1 a k a j a k j +...+a k a 0 )x k (k + 1)a k+1 = a 0 a k + a 1 a k a j a k j a k a 0, k = 0, 1,... 30

31 a 1 = a 2 0, 2a 2 = a 0 a 1 + a 0 a 1 = 2a 3 0, a 2 = a 3 0 3a 3 = a 0 a 2 + a 1 a 1 + a 2 a 0 = 3a 4 0, a 3 = a a k = a k+1 0, k = 1,... a 0 = c a k = c k+1 y = c n+1 x n = c(1 + cx + c 2 x c k x k +..) = n=0 cx < 1 c ( cx < 1) 1 cx x < 1 c c 1 cx y = y 2 y = y 2 dy y 2 = dx y = 1 x + C = 1 y = x + C, 1 C(1 ( x C )) = x 2 y = y x(x = 0, y = 0) x(x 1)y + (3x 1)y + y = 0((x = 0, y = a) c 1 cx (c = 1 C ) 31

32 2.4 y = u u xy 2y = x 3, y = u u 2 x u = x2, u = x 3 + Cx 2 y = x 3 + Cx 2 y = x4 4 + Cx3 + D y = p p = dy dx, y = dp { yy +2(y ) 2 = yy yp dp dy +2p2 = yp 0 y = 0 y = C(const.) (ii) dp dy + 2p y dx = dp dy dy dx = p dp dy (i)p = 0 (ii) dp dy + 2p y = 1, (i)p = = 1 p = 1 3 y + C y 2 y = 1 3 y + C y 2 = y3 +C 3y 2 3y 2 dy y 3 +C = dx log(y3 + C) = x + D y + P (x)y + Q(x)y = R(x) 32

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y

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