Captulo

2

y x f x, y(x), Dy(x), D 2 y(x), . . . , Dn y(x) = r(x)

Dy(x), D2 y(x), . . . , Dn y(x) 1, 2, . . . , n y(x)

n F (x, y, C1 , C2 , . . . , Cn )

y (x) xy(x) = 0 a2 y (t) + a1 y (t) + a0 y(t) = f (t) y (t) + 4 sin(y(t)) = 0 x3 y (x) + x2 y (x) + xy (x) + y(x) = f (x) x2 y (x) + xy (x) + (x2 2 )y(x) = 0

F (x, y, C) = 0 F (x, y, C1 , C2 , . . . , Cn ) = 0 n n

2xy a bx = 0 a, b

a = 1, b = 1 a = 3, b = 1

2xy + 2y b = 0

2xy + 4y = 0

y +

2y =0x

y = p

dp 2pdp 2dx+=0+=0dxxpx

ln(p) = 2 ln(x) = ln(C1 ) px2 = C1

y y = C1 x1 + C2

y = C1 ex + C2 e2x + x

y = C1 ex 2C2 e2x + 1y = C1 ex + 4C2 e2x

C1yx=y 1C2

exe2xex 2e2x

yxe2x y 1 2e2xC1 = xe2x e x e2e2x

;

x eyx ex y 1 C2 = x2xeex2x e2e

C1 = ex (2y 2x + y 1)C2 = e2x (y + x y + 1)

y = 2y 2x + y 1 + 4(y + x y + 1)

y + 3y + 2y = 2x + 3

y=x ,

y = ex + x

y1 (x) y2 (x) yss (x) I Ry = C1 y1 (x) + C2 y2 (x) + yss (x)

y = C1 y1 + C2 y2 + yssy = C1 y1 + C2 y2 + yss

y1 y2y1 y2

C1y yss=C2y yss

1(y y y2 yss y2 y + y2 yss)W (x) 21(y1 y + y1 yss + y1 y y1 yss)C2 =W (x)y2 y2 C1 =

y W (x) = 1y1 y1 y2 I

y = y1

11) + y2) + yss(y2 y y2 yss y2 y + y2 yss(y1 y + y1 yss + y1 y y1 yssW (x)W (x)

y

11(y1 y2 y2 y1 )y +(y y y2 y1 )y = r(x)W (x)W (x) 1 2

r(x) r(x) = yss

11(y1 y2 y2 y1 )yss(y y y2 y1 )yss+W (x)W (x) 1 2

y + p(x)y + q(x)y = r(x)

y1 y2 y1 y2p(x) =W (x)

y1 y2 y1 y2q(x) =W (x)

,

y = C1 x + C2 ex + x2

x exW (x) = 1 ex

p(x) q(x)

x ex 0 ex x=p(x) =xe (x + 1)x+1

,

= ex (x + 1) 1 ex 0 ex 1q(x) ==xe (x + 1)x+1

r(x) = 2 +

2(x + 1) + 2x2 x212 + 2x + x2x2x x2 ==x+1x+1x+1x+1

y +

x2 + 2x + 2x 1y y=x+1x+1x+1

(x + 1)y + xy y = x2 + 2x + 2

y1 y2 W (x) = y1 y2 y2 y1 W (x) = y1 y2 y2 y1

W (x) p(x) = W (x)

dW (x)= p(x)dxW (x)

W (x) = Ke

p(x)dx

K W (x) = Ke

xdxx+1

= Kex+ln(x+1) = K(x + 1)ex K = 1

n y1 , y2 , y3 . . . yn I y c = C 1 y 1 + C 2 y 2 + C 3 y 3 + + Cn y n

C1 , C2 , C3 . . . , Cn I R C1 y1 (x) + C2 y2 (x) + + Cn yn (x) 0

x2 , x2 , . . . , xn n n

y1 (x1 ) y2 (x1 ) yn (x1 ) y1 (x2 ) y2 (x2 ) yn (x2 )y1 (xn ) y2 (xn ) yn (xn )

C1 C2 = Cn

00 0

R

{1, x, x2 }

C1 +C2x+C3x2 0

x = 1 C1 C2 + C3 = 0

x = 1 C1 + C2 + C3 = 0

x = 2 C1 + 2C2 + 4C3 = 0

1 1 1 1 1 1 =6 1 2 4

n {y1 , y2 , . . . , yn } x0 I y1 (x0 )y2 (x0 ) yn (x0 ) y (x0 )y2 (x0 ) yn (x0 ) 1W (x0 ) = n1 y1 (x0 ) y2n1 (x0 ) ynn1 (x0 )

n 1

{x, ex } R

x exW (x0 ) = 1 ex

= ex (x + 1)

y1 , y2 , y3 . . . yn I

C1 y1 (x) + C2 y2 (x) + C3 y3 (x) + + Cn yn (x) 0

y2 (x0 ) yn (x0 )y1 (x0 ) y (x0 )y2 (x0 ) yn (x0 ) 1n1n1n1y1 (x0 ) y2 (x0 ) yn (x0 )

C1 C2 = Cn

Ci =

00 0

0W (x0 )

x0

{2x2 , |x|x} (, 0) (0, ) R

f (x) = 2x2 g(x) = x|x|

g(x) =

x2x2

x < 0 x 0

(, 0) (0, ) (, 0) f (x)/g(x) = 2 (0, ) f (x)/g(x) = 2 f (x), g(x)

(, 0) (0, ) 2 2xW (x) = 4x 2 2xW (x) = 4x

x2 = 4x3 + 4x3 = 0 < x < 02x x2 = 4x3 4x3 = 0 0 x < 2x

R C1 x2 + C2 x|x| 0

x = 1

1 1 1 1

1 11 1

C10=0C2

= 2 = 0 R

y + p(x)y + q(x)y = r(x)

D D2 + p(x)D + q(x) y = r(x)

L2 (x, D)y = D 2 + p(x)D + q(x)

L2 (x, D)y = r(x)

y + p(x)y + q(x)y = 0

L2 (x, D)y = 0

y = C1 y1 + C2 y2 + yss

y = yc + yss

y1 , y2 I yc = C 1 y1 + C2 y2

{y1 , y2 } CF S I

y1 , y2 W (x) = 0 x I

y1 , y2 L2 (x, D)y1 0L2 (x, D)y2 0

C1 L2 (x, D)y1 0 L2 (x, D)C1 y1 0C2 L2 (x, D)y2 0 L2 (x, D)C1 y2 0

L2 (x, D)[C1 y1 + C2 y2 ] 0

yss y = yc + yss = C1 y1 + C2 y2 + yss

y + p(x)y + q(x)y = r(x) ,

y(x0 ) = y0

,

y (x0 ) = p0

(x0 , y0 ) p0 y(x) = C1 y1 (x) + C2 y2 (x) + yss (x)(x)y (x) = C1 y1 (x) + C2 y2 (x) + yss

(x0 , y0 )

y1 (x0 ) y2 (x0 )y1 (x0 ) y2 (x0 )

C1y0 yss (x0 )=C2(x0 )p0 yss

p(x) q(x) r(x)

(x + 1)y + xy y = x2 + 2x + 2 ,

y(1) = 1 ,

y (1) = 1

y(x) = C1 x + C2 ex + x2

x2 + 2x + 2

1

x

q(x) = r(x) = p(x) =x+1x+1x+1 (1, )

1 e11 e11

C10=1C2

e 1.36 211y(x) = x + ex+1 + x222

, C1 = 2

C2 =

y1 (x) (x) y = y1 (x)(x)

y = y1 + y1 y = y1 + 2y1 + y1

y1 + 2y1 + y1 + p(x)[y1 + y1 ] + q(x)y1 r(x)

y1 + [2y1 + y1 p(x)] + [y1 + p(x)y1 + q(x)y1 ] r(x)

y1 y1 + [2y1 + y1 p(x)] = r(x)

+

2y1r(x)+ p(x) =y1y1

= z dz+dx

2y1r(x)+ p(x) z =y1y1

(x) = y12 e

p(x)dx

z z = A

1

+

1

r(x)dxy1

A (x) = B + A

1

dx +

1

r(x)dxdxy1

y1

1 dxr(x)1dxdxyss = y1 y1y2 = y1

y = x x2 y + xy y = x

p(x) = x1

= y12 e

p(x)dx

= x2 e

1dxx

= x3

y2 = x

1x3 dx = x12

{y1 , y2 } = {x, x1 }

1x

r(x) = yss = x

x

3

1x3 x1dxdx = x ln(x)x2

2 x log (x) x+ x 42xx 4 x ( 2) y =

y = x

(x + 1)y + xy y = 0

p(x) = x +x 1

= y12 e

p(x)dx

= x2 e

=

xdxx+1

x2 e xx+1

y2 = x

ex (x + 1)dx = exx2

yc = C1 x + C2 ex

eax (x + 1)a2 eax + xaeax eax = 0 eax (a2 x + a2 + ax 1) = (a + 1)[xa + a 1] = 0

a = 1 y1 = ex xm (x + 1)m(m 1)xm2 + xmxm1 xm = 0 [m(m 1)x1 + m(m 1)x2 + m 1]xm = (m 1)[mx1 + mx2 + 1] = 0

m = 1 y2 = x

(x2 + 2x)y + (x2 2)y 2(x + 1)y = 0

y = eax

(x2 + 2x)a2 eax + (x2 2)aeax 2(x + 1)eax = 0

(a2 + a)x2 + (a2 1)2x 2(a + 1) = 0

a = 1 y1 = ex p(x) =

x2 22(x + 1) q(x) = x(x + 2)x(x + 2)

1

1

p(x) p(x) = 1 x x+2 = e2x e

1)dx(1 x1 x+2

= e2x

exex=x(x + 2)x(x + 2)

y2 = e

x

x(x + 2)ex dx = ex x2 ex = x2

y = c1 ex + c2 x2

{sin(x), cos(x)}

{ex , ex }

{x2 , ex }

f (x) I R {f (x), xf (x)} I {1, cos(2x), sin2 (x)} {x2 , x|x|} y = C1 ex + C2 e2x y = C1 ex + C2 xex y = C1 ex cos(x) + C2 ex sin(x) y = C1 cos(x) + C2 sin(x) + ex

y + 2y + 2 y = 0 y1 = ex y2 =xex

x2 y + xy 4y = 3x y = x2 y(1) = 1 y (1) = 1

x2 y xy + y = 0 y = x

y(1) = 1 y (1) = 0

xy + 2y xy = x y = x1 ex y(1) = 1 y (1) = 1

xy + 2y + xy = x y = x1 sin(x) y(1) = 1 y (1) = 1

an Dn + an1 Dn1 + + a1 D + a0 y(x) = r(x)

an Dn + an1 Dn1 + + a1 D + a0 y(x) = 0

y = yc + yss

y c = C 1 y 1 + C 2 y 2 + + Cn y n

ex n n an n + an1 n1 + + a1 + a0 = 0

a2 D2 + a1 D + a0 y(x) = 0

D2 + pD + q y(x) = 0

2 + p + q = 0

1 , 2 =

p

p2 4q2

p2 4q > 0 CF S = {y1 , y2 } = {e1 x , e2 x }

p2 4q = 0 1 =p2 = = 2

CF S = {y1 , y2 } = CF S = {ex , xex }

p2 4q < 0 1 , 2 = j p

4q p2

= =22 j j = 1 CF S = {y1 , y2 } = {e(+j)x , xe(j)x }

ej ej = cos() j sin()

CF S = {y1 , y2 } = {ex [cos(x) + j sin(x)] , ex [cos(x) j sin(x)]}

{y1 , y2 } {a(y1 + y2 ), b(y1 y2 )} a, b

CF S = {y1 , y2 } = {ex cos(x), ex sin(x)}

(D2 + 3D + 2)y(x) = 0

(D2 + 2D + 2)y(x) = 0

(D2 + 2D + 1)y(x) = 0

(D2 + 4)y(x) = 0

2 + 3 + 2 = 0 1 = 1 2 = 2 CF S = {ex , e2x }

2 + 2 + 1 = 0 1 = 1 2 = 1 CF S = {ex , xex } 2 + 2 + 2 = 0 1 , 2 = 1 j1 CF S = {ex cos(x), ex sin(x)}

2 + 4 = 0 1 , 2 = 0 j2 CF S = {cos(2x), sin(2x)}

D a2 2 + a1 + a0 = 0

n n Ln (D)y(x) = 0

Ln () = 0 n n 1 , 2 , 3 CF S = {e1 x , e2 x , e3 x }

1 = 2 = , 3 CF S = {ex , xex , e3 x } 1 = 2 = 3 = CF S = {ex , xex , x2 ex } 1 , 2,3 = j

CF S = {e1 x , ex cos(x), ex sin(x)}

(D3 + 4D2 + 4D)y(x) = 0

(D4 16)y(x) = 0

(D3 + D2 4D 4)y(x) = 0

(D4 + 2D3 + 3D2 + 2D + 2)y(x) = 0

(D3 + 3D2 + 3D + 1)y(x) = 0

(D4 + 5D2 + 6)y(x) = 0

(D3 + 8)y(x) = 0

(D5 + 4D3 )y(x) = 0

(D4 + 4D2 + 3)y(x) = 0

(D4 + D2 + 1)y(x) = 0

3 + 42 + 4 = 0 (2 + 4 + 4) = 0 ( + 2)2 = 0 CF S = {1, e2x , xe2x } 3 + 2 4 4 = 0 2 ( + 1) 4( + 1) = 0 (2 4)( + 1) = 0

CF S = {e2x , e2x , ex }

3 + 32 + 3 + 1 = 0 ( + 1)3 = 0 CF S = {ex , xex , x2 ex } 3 + 8 = 0 ( + 2)(2 2 + 4)= 0 CF S = {e2x , ex cos( 3 x), ex sin( 3 x)}

4 + 42 + 3 = 0 (2 + 1)(2 + 3) = 0 CF S = {cos(x), sin(x), cos( 3 x), sin( 3 x)} 4 16 = 0 (2 4)(2 + 4) = 0 CF S = {e2x , e2x , cos(2x), sin(2x)} 4 + 23 + 32 + 2 + 2 = 0

( 1) (l2 + 1)(l2 + 2l + 2) = 0

(2 + 1)(2 + 2 + 2) = 0 CF S = {cos(x), sin(x), ex cos(x), ex sin(x)} 4 + 52 + 6 = 0

( 1) (l2 + 2)(l2 + 3) = 0

CF S = {cos( 2 x), sin( 2 x), cos( 3 x), sin( 3 x)}

5 + 43 = 0 3 (2 + 4) = 0 CF S = {1, x, x2 , cos(2x), sin(2x)} 4 + 2 + 1 = 0

( 1) (l2 l + 1)(l2 + l + 1) = 0

xxxxCF S = {e 2 cos( 3 x/2), e 2 sin( 3 x/2), e 2 cos( 3 x/2), e 2 sin( 3 x/2)}

(D3 + 4D2 + 3D)y(x) = 0

(D4 + 4)y(x) = 0

(D4 + D3 4D2 4D)y(x) = 0

(D4 + 2D3 + D2 2D 2)y(x) = 0

(D3 + 2D2 + 2D + 1)y(x) = 0

(D4 5D2 + 6)y(x) = 0

(D4 + 8D)y(x) = 0

(D5 + 8D2 )y(x) = 0

(D4 4D2 + 3)y(x) = 0

(D4 + 2D2 + 1)y(x) = 0

(an Dn + an1 Dn1 + + a1 D + a0 )y(x) = f (x)

Ln (D)y(x) = f (x)

m1 Ln (D) Ln (D) = Ln1 (D)[D m1 ]

Ln1 (D)[D m1 ]y(x) = f (x)

1 (x) = [D m1 ]y(x) n 1 Ln1 1 (x) = f (x)

m2 Ln1 (D) Ln2 (D)[D m2 ]1 (x) = f (x)

2 (x) = [D m2 ]1 (x) n (a2 D2 + a1 D + a0 )y(x) = f (x)

(D2 + pD + q)y(x) = r(x)

(D m2 )(D m1 )y(x) = r(x)

1 (x) = (D m1 )y(x) (D m1 )y(x) = 1 (x)(D m2 )1 (x) = r(x)

(D2 1)y(x) = x

yc = C1 ex + C2 ex

(D + 1)(D 1)y(x) = x

(D 1)y(x) = 1 (x)(D + 1)1 (x) = x

= ex x

e 1 = C 1 +

xex dx

1 = C1 ex + x 1

(D 1)y(x) = C1 ex + x 1

= ex x

e y(x) = C2 +

ex (C1 ex + x 1)dx

1y(x) = C2 ex C1 ex x2

yss = x y(x) = C1 ex + C2 ex x

(D2 1)y(x) = 2 sin(x)

yc = C1 ex + C2 ex

(D 1)y(x) = 1 (x)(D + 1)1 (x) = 2 sin(x)

= ex x

e 1 = C 1 +

2ex sin(x)dx

1 = C1 ex + sin(x) cos(x)

1 = sin(x) cos(x)

(D 1)y(x) = sin(x) cos(x)

yss = e

x

ex (sin(x) cos(x))dx = sin(x)

yc = C1 ex + C2 ex sin(x)

n Ln (D)y(x) = f (x)

f (x) m {f1 , f2 , . . . , fm }

f (x) f (x) yss = A1 f1 + A2 f2 + + Am fm

Ln (yss ) f (x)

{f1 , f2 , . . . , fm } f (x) xm m = 1, 2, 3, . . . yss = xm (A1 f1 + A2 f2 + + Am fm )

f (x) f (x) = xm eax

cos(bx)sin(bx)

m 0

(D2 1)y(x) = x

CF S = {ex , ex }

f (x) = x {x, 1} yss = A1 x + A2

Dyss = A1

,

D2 yss = 0

0 (A1 x + A2 ) x A1 = 1, A2 = 0 yss = x

(D2 + 3D + 2)y(x) = sin(x)

CF S = {ex, e2x}

f (x) = sin(x) {sin(x), cos(x)} yss = A1 sin(x) + A2 cos(x)

(D2 + 3D + 2)[A1 sin(x) + A2 cos(x)] sin(x)

A1 ( sin(x) + 3 cos(x) + 2 sin(x)) + A2 ( cos(x) 3 sin(x) + 2 cos(x)) sin(x)

A1 (sin(x) + 3 cos(x)) + A2 (3 sin(x) + cos(x)) sin(x)

A1 3A2 = 13A1 + A2 = 0

yss =

31sin(x) cos(x)1010

(D3 + D)y(x) = x2 ex

CF S = {1, cos(x), sin(x)}

f (x) = x2 ex {x2 ex , xex , ex } yss = A1 x2 ex + A2 xex + A3 ex

y (x) =

2cos (x) 2 ddx2 y (x)

x=0

+1

2 dsin (x) 2 d x y (x)x=0 1+22 x5 exx e 2 x ex ++ y (0) 2+22

d2y(x)+d x2x=0

A1 = 12 A2 = 2 A3 = 52

(D2 + 3D + 2)y(x) = x2 ex

CF S = {ex , e2x }

{x2 ex , xex , ex } x yss = x A1 x2 ex + A2 xex + A3 ex

yss =

ex 3x 3x2 + 6x3

y =

(x3 3 x2 +6 x6) ex3

+ %k1 ex + %k2 e2 x

(D3 + 4D)y(x) = 2 sin(2x) + 5 cos(2x)

CF S

CF S = {1, cos(2x), sin(2x)}

{sin(2x), cos(2x)} yss = x [A1 cos(2x) + A2 sin(2x)]

(D3 + 4D2 )y(x) = x3 + 6x + 4

CF S

CF S = {1, x, e4x }

{x3 , x2 , x, 1} yss = x2 A1 x3 + A2 x2 + A3 x + A4

e4 x

y (x) =e4 xx5x4 772048+ 80 64+

d2d x2

y(x)

x=0

16217 x3+ 77256x64

x

d2d x2

y(x)

d2d x2

x=0+477 x77 512 + y (0) + 2048

y(x)16

x=0

+x

ddx

y (x)x=0

77 A1 = 801 , A2 = 641 , A3 = 17, A4 = 25664

Ln (D)y(x) = f1 (x) + f2 (x) f1 , f2 yss = yss1 + yss2

yss1 Ln (D)y(x) = f1 (x) yss2 Ln (D)y(x) = f2 (x)

(D3 + 4D2 )y(x) = x3 + 6e4x

CF S

CF S = {1, x, e4x }

x3 {x3 , x2 , x, 1} yss1 = x2 A1 x3 + A2 x2 + A3 x + A4

6e4x yss2 = x A5 e4x

yss = x2 A1 x3 + A2 x2 + A3 x + A4 + A5 xe4x

Ln (D)y(x) = f (x)

yss =

1f (x)Ln (D)

1

f (x) D 1f (x) =D

f (x)dx

1f (x) k = 1, 2, 3 . . . Dk111 11f (x)dxdx = f (x)dx dxf (x) = k1 f (x) = k1 f (x)dx = k2DkDDDD

k

k 1 xD f (x) =(x t)p1 f (t)dt (p) 0

p

(D m)y(x) = f (x)

y = c1 e

mx

+e

mx

emx f (x)dx

1yss =f (x) = emxDm

emx f (x)dx1

Dm f (x) 1f (x) = emxDm

emx f (x)dx

(D 2)y(x) = sin(x)

1yss =sin(x) = e2xD2

e2x sin(x)dx

21yss = cos(x) sin(x)55

1 k(D m)k

1111mxmxf (x) =ef (x) =f (x)dxe(D m)k(D m)k1 D m(D m)k1

k = 211mxmxmxeemx f (x)dxdxef (x) =f (x)dx = e2(D m)Dm

(D2 + 4D + 4)y(x) = e2x

1e2x = e2xyss =(D + 2)2

1e2x e2x dxdx = x2 e2x2

(D m)y(x) = eax

1eaxDm

yss =

yss = e

mx

e

mx ax

e dx = e

mx

e

(am)x

dx = e

11eax =eaxDmam

mx e

(am)x

am

=

1eaxam

a = m

a = m

D a m a = m 1eax = eaxDa

eax eax dx = xeax

n Ln (D)y(x) = eax

yss =

1eaxLn (D)

yss =

1eax(D m1 )(D m2 ) (D mn )

A1A2Aneax++ +yss =D m1 D m2D mn

a = mk

A1A2An++ +eaxa m1 a m2a mn Ln (D)y(x) = eax yss =

1eax Ln (a) = 0Ln (a) Ln (a) = 0 eax yss =

1eax(D a)Ln1 (D) Ln1 (a) = 0 ax 11e1axe ==xeaxyss =(D a)Ln1 (D)Ln1 (a) D aLn1 (a)yss =

Ln (D) = (D a)Ln1 (D) Ln (D) = (D a)Ln1 (D) + Ln1 (D)

D = a Ln (a) = Ln1 (a) Ln (D)y(x) = eax 1eaxL(a)n 1 xeaxyss = Ln (a)1 n xn eaxLn (a)

Ln (a) = 0

Ln (a) = 0 Lnn (a) = 0

(D3 + 2D2 + 3D + 2)y(x) = e2x

(D3 + 2D2 + 3D + 2)y(x) = ex

L3(2) = (8 + 8 6 + 2) = 4

1yss = e2x4 L3 (1) = 1+23+2 = 0

L3 (D) = 3D 2 + 4D + 3

D = 1 yss =

11xex = xex34+32

a = 0 Ln (D)y(x) = E 1EL(a)n 1xEyss = Ln (a)1 n xn ELn (a)

Ln (a) = 0

Ln (a) = 0 Lnn (a) = 0

Ln (D)y(x) =

sin(x)cos(x)

Ln (D)y(x) = cos(x) = yss =

e+jx + ejx 2

11e+jx +ejx2Ln (+j)2Ln (j)

D = j D2 = 2 jx1 1 e + ejxyss =cos(x)=Ln (D) D2 =22Ln (D) D2 =2

Ln ( 2 ) = 0

1 cos(x) Ln ( 2 ) = 0L(D)22nD =1xcos(x) Ln ( 2 ) = 0Ln (D) D2 =2yss = n1cos(x) Lnn ( 2 ) = 0xLn (D) 22n

D =

(D2 + 3D + 2)y(x) = sin(x)

11sin(x)sin(x) =yss = 2D + 3D + 2 D2 =13D + 1

13D 1 3D 1sin(x) = [3 cos(x) sin(x)]sin(x) =yss =29D 1 D2 =11010

yss =

31sin(x) cos(x)1010

(D3 + 4D)y(x) = 2 sin(2x) + 5 cos(2x)

L3(D2) = 0 D2 = 4

1yss =x [2 sin(2x) + 5 cos(2x)]3D2 + 4 D2 =4

1yss = [2x sin(2x) + 5x cos(x)]8

Ln (D)y(x) = xm m = 0, 1, 2, 3, . . . yss =

11xm =xm2Ln (D)a0 + a1 D + a2 D + + an1 Dn1 + an Dn

11xmyss =a0 1 + (D)

(D) =

a1 D + a2 D2 + + an1 Dn1 + an Dna0

11 a0 1 + (D)

1 1 k1123mm+1=1 (D) + (D) (D) + + (D) + (D) + = (D)xma0 1 + (D)a0a0 k=0

m+1 (D), m+2 (D), . . . xm yss =

m1 1 k1 (D) + 2 (D) 3 (D) + + m (D) xm == (D)xma0a0 k=0

(D3 + 4D2 )y(x) = x3

111 33yss = 3x =xD + 4D2D+4D2

x3

1yss =D+4

x520

11D D2 D3D4D515=x =1 ++x580 1 + D/48041664256 1024

12015x4 20x3 60x2 120x5x ++yss =80416642561024

3 + 42 = 0 CF S = {1, x, e4x }

yss =

1 5113 2x x4 + x3 x806464256

1yss = 2D

1D+4

1x =4D23

1yss =4D2

1yss =4

11 + D/4

1x =4D23

63x2 6x+x 416 643

x5 x4 x3 3x2++20 64 1664

=

D D2 D31 +41664

x3

x5x4x3 3x2++80 256 64 256

Dy(x) = eax f (x)

1yss = [eax f (x)] =D

1f (x) = eaxD+a

eax f (x)dxf (x)D+a

eax f (x)dx

eax f (x)dx = eax

1f (x)D+a

1 ax1[e f (x)] = eaxf (x)DD+a

Ln (D)y(x) = eax f (x) yss =

11[eax f (x)] = eaxf (x)Ln (D)Ln (D + a)

cos(x)f (x)dx sin(x)f (x)dx

e

jx

f (x)dx =

cos(x)f (x)dx + j

sin(x)f (x)dx

f (x)f (x)(D j)f (x)ejx f (x)dx =ejx= ejx=[cos(x)+jsin(x)(Dj)]D + jD2 + 2D2 + 2Df (x)Df (x)f (x)f (x)+ j sin(x) 2+ sin(x) 2 cos(x) 2= cos(x) 2D + 2D + 2D + 2D + 2

Df (x)f (x)f (x)()cos(x) 2=Dcos(x) 2+ sin(x) 2D + 2D + 2D + 2

D()

Re

e

jx

f (x)dx = D

()

f (x)cos(x) 2D + 2

+ jD

()

ejx f (x)dx = cos(x)dx

cos(x)f (x)dx = D

()

f (x)sin(x) 2D + 2

f (x)cos(x) 2D + 2

Im

11 + D/a

ejx f (x)dx = sin(x)dx

eax x2 dx

1 ax 2 1eaxe x = eaxx2 =DD+aa

eaxx =a2

D D21 + 2aa

x2

eaxe x dx =aax 2

22xx + 2aa2

eax cos(bx)dx

11 axaxax D acos(bx)(e cos(bx)) = ecos(bx) = e 2 2DD+aD2 a2D =b

eaxDa1 ax(e cos(bx)) = eax 2cos(bx)=[b sin(bx) a cos(bx)]Db a2a2 + b 2

eax cos(bx)dx =

(D2 + 3D + 2)y(x) = xex

eax[b sin(bx) + a cos(bx)]a2 + b 2

2 + 32 + 2 = 0

CF S = {ex , e2x }

x2xyss = x A1 xe

+ A2 e

yss =

D2

111ex x = exx = ex 2x2+ 3D + 2(D 1) + 3(D 1) + 2D +D

yss = ex

x2111x = ex (1 D) x = ex (x 1) = ex ( x)D(D + 1)DD2 x2 sin(bx)dx

x2 sin(bx)dx = D()

2 2 Dx2xsin(bx) 2= D() sin(bx) 1 2D +bbb2

x22sin(bx) 2 4x sin(bx)dx = Dbb 2x22x= sin(bx) 2 b cos(bx) 2 4bbb 2x22x 3 cos(bx)= 2 sin(bx) bbb2

()

x2ex cos(x)dx2 x f (x) = x e

2 x

xe

cos(x)dx = D

()

(x2 ex )cos(x) 2D +1

= cos(x)

D(x2 ex )(x2 ex )+sin(x)D2 + 1D2 + 1

(x2 ex ) D2 + 12x2ex(x2 ex )exxx=e=1=2D2 + 1(D 1)2 + 1221 + D 2D2x ex(24x)e=x2 +2 =x2 2x + 1222

D 2 2D2

+

D 2 2D2

2

x2

ex exex 2D(x2 ex )2x+1==(2x2)x1 x22D +1222

x2 ex cos(x)dx =

ex (1 x2 ) cos(x) + (x2 + 2x + 1) sin(x)2

Eeax cos(bx)Ln (D)y(x) =sin(bx)nAn x + An1 xn1 + + A1 x + A0

{y1 , y2 , . . . , yn } y (x) + p(x)y + q(x)y = r(x)

{y1 , y2 } 1 , 2 yss = y1 1 + y2 2

= y1 1 + y1 1 + y2 2 + y2 2 yss= y1 1 + y2 2 + y1 1 + y2 2yss

= y1 1 + y1 1 + y2 2 + y2 2 + y1 1 + y1 1 + y2 2 + y2 2yss

y1 1 + y1 1 + y2 2 + y2 2 + y1 1 + y1 1 + y2 2 + y2 2 +p(x)[y1 1 + y2 2 + y1 1 + y2 2 ]+q(x)[y1 1 + y2 2 ] r(x)

[y1 + p(x)y1 + q(x)]1 + [y2 + p(x)y2 + q(x)]2 +2(y1 1 + y2 2 ) + p(x)[y1 1 + y2 2 ] + y1 1 + y2 2 r(x)

y1 , y2 2(y1 1 + y2 2 ) + p(x)[y1 1 + y2 2 ] + y1 1 + y2 2 r(x)

p(x) y1 1 + y2 2 = 0 y1 1 + y2 2 + y1 1 + y2 2 = 0

y1 1 + y2 2 = r(x)

1 , 2 y1 y1y2 y2

12

=

0r(x)

0 y2 r(x) y21 =W (x)

y10 y1 r(x)2 =W (x)

dn ydn1 ydn2 ydy+p(x)+p(x)+ + pn1 (x) + pn (x)y = r(x)12nn1n2dxdxdxdx

{y1 , y2 , . . . , yn } yss = y1 1 + y2 2 + + yn n

y 2 yny1 yy2 yn 1 y1n1 y2n1 ynn1

10 0 2 = r(x)n

(D2 + 3D + 2)y(x) = xex

CF S

CF S = {ex , e2x }

yss = 1 ex + 2 e2x

ex exe2x 2e2x

10=xex2

1 = x 2 = xex 1 1 = x22

,

2 = (x + 1)ex

yss =

1 2 x1x e + ((x + 1)ex ) e2x = x2 ex xex + ex22

1yss = x2 ex xex2

(D2 + 4)y(x) = sec(2x)

CF S

CF S = {cos(2x), sin(2x)}

yss = 1 cos(2x) + 2 sin(2x)

cos(2x)sin(2x)2 sin(2x) 2 cos(2x)

10=sec(2x)2

12

12

1 = tan(2x) 2 = 1 =

1ln (cos(2x))4

,

12 = x2

yss =

11ln (cos(2x)) cos(2x) +x sin(2x)42

(D3 + D)y(x) = tan(x)

CF S CF S = {1, cos(x), sin(x)}

yss = 1 + 2 cos(x) + 3 sin(x)

11 cos(x)sin(x)0 0 sin(x) cos(x) 2 = 00 cos(x) sin(x)tan(x)3

1

= tan(x)

2

= sin(x)

3

11 = ln(cos(x)) ,2

2 = cos(x) ,

sin2 (x)=cos(x)

3 = sin(x) ln(sec(x) + tan(x))

1yss = ln(cos(x)) + cos2 (x) + [sin(x) ln(sec(x) + tan(x))] sin(x)2

sin2 (x) + cos2 (x) = 1 1yss = ln(cos(x)) sin(x) ln(sec(x) + tan(x))2

(D3 + 3D2 )y(x) = x2 + xe3x (D2 + 4)y(x) = x cos2 (x) (D4 +2D2 +1)y(x) = sin(x) sin(2x) (D3 + 3D2 + 2D)y(x) = x2 + sinh(x)

(D3 + 2D2 + 2D + 1)y(x) = x cosh(x) (D3 + D2 + D + 1)y(x) = x2 + x sin(x) + xex (D4 + 5D2 + 4)y(x) = x sin(x) + x2 cos(2x) (D3 + 4D2 + 6D + 4)y(x) = xex sin(x) (4D2 + 4D + 1)y(x) = xe 2

x

(D4 + 2D3 + 5D2 + 8D + 4)y(x) = xex + x sin(2x) (D3 + 3D2 )y(x) = x2 + xe3x (D2 + 4)y(x) = cos2 (x) (D4 + 2D2 + 1)y(x) = sin(x) (D3 + 3D2 + 2D)y(x) = x2 + ex (D3 + 2D2 + 2D + 1)y(x) = xex (D3 + D2 + D + 1)y(x) = x2 + sin(x) (D4 + 5D2 + 4)y(x) = ex sin(x) (D3 + 4D2 + 6D + 4)y(x) = ex sin(x) (4D2 + 4D + 1)y(x) = xe 2

x

(D4 + 2D3 + 5D2 + 8D + 4)y(x) = xex + x sin(2x) (D2 + 1)y(x) = csc(x) (D2 + 2D + 1)y(x) = ex sec(x) (D2 + 4D + 4)y(x) = e2x ln(x) (D2 + D)y(x) = ex x1 (D3 + 4D)y(x) = cot(2x)

n t x1 (t), x2 (t), . . . , xn (t) n dx1 = f1 (t, x1 , x2 , . . . , xn )dxdx2 = f2 (t, x1 , x2 , . . . , xn )dx

dxn = fn (t, x1 , x2 , . . . , xn )dx

dx1 = a11 x1 + a12 x1 + + a1n xn + r1 (t)dxdx2 = a21 x1 + a22 x1 + + a2n xn + r2 (t)dx

dx1 = an1 x1 + an2 x1 + + ann xn + rn (t)dx

x1 d x2 =dt xn

a11 a12 a1na21 a22 a2n an1 an2 ann

x1r1 (t)x2 r2 (t) + rn (t)xn

dX(t) = A X(t) + r(t)dt

X(t) A r(t)

dx(t) = ax(t) + br(t) dtd x(t) ax(t) = br(t) = eat dt atatx(t) = Ce + eeat r(t)dt

x(0) at

x(t) = x(0)e + e

at

t

ea r( )d

0

X(0) tX(t) = X(0)eAt + eAt

eA r( )d

0

= eAt eAt = I + At +

A2 2 A3 3t +t + 2!3!

I eAt Xc (t) = c1 V1 e1 t + c2 V2 e2 t + + cn Vn en t

1 , 2 , . . . , n A V1 , V2 , . . . , Vn c1 , c2 , . . . , cn Xss (t) X = r(t) X

Vn en t Xss (t) = X

ddt

x1x2

=

1 11 1

t x1e+x2et

A

1 1p() = |A I| = 11

= (1 + )2 1 = 2 + 2 = ( + 2) = 0

1 = 0 2 = 2 CF S = {1, e2t } (A I) V = 0 1 = 0 v1 + v2 = 0 v1 v2 = 0 11

V1 =

1 = 2 v1 + v2 = 0 v1 + v2 = 0 11

V2 =

V1 =

11

,

V2 =

11

Xc (t) = c1

11

0t

e + c2

11

e2t

X=

1 e2t1 e2t

1 e2t1 e2t

t 1e=2et

e3t et1 t1 tttee=+e=+e12et22tte e1 t1 t e3t3t=e e 2 =e 2 =2et223

1 =

Xss (t) =

1 e2t1 e2t

ete e1 t e3t = t 3 t 2e3e2e 33

t

t

X(t) = c1

11

+ c2

11

r(t)

e

2t

1+3

ett2e 3et

A

1 111 teet

e2 t 1 2 +2 Xc = 1e2 t22

1e2 t22 e2 t 1 +22

{1, e2t }

eAt r(t) eAt

et (2 e2 t 3)3 Xss = et3

1 = 0, 2 = 2 1 1 [1, 1] [1, 1] V2 V1 1 2

X

et3

et (2 e2t 3) 3

A r(t) t

eAt

{1, e2t }

11 V1 = 1211V2 = 12

1 , 2

X

D L11 (D)x1 + L12 (D)x2 + + L1n (D)xn = r1 (t)L21 (D)x1 + L22 (D)x2 + + L2n (D)xn = r2 (t)

Ln1 (D)x1 + Ln2 (D)x2 + + Lnn (D)xn = rn (t)

L11 L12 L1n L21 L22 L2n Ln1 Ln2 Lnn

x1r1 (t) x2 r2 (t) = rn (t)xn

L11 L12 L1n L21 L22 L2nLn (D) = Ln1 Ln2 Lnn

n Ln (D)x1 (t) = f1 (t)

r(t) r1 (t) L12 L1n r2 (t) L22 L2nf1 (t) = rn (t) Ln2 Lnn

L (D) L12 (D)L2 (D) = 11L21 (D) L22 (D)

r (t) L12 (D)L2 (D)x1 (t) = 1r2 (t) L22 (D) L (D) r1 (t)L2 (D)x2 (t) = 11L21 (D) r2 (t)

= L22 (D)r1 (t) L12 (D)r2 (t) = L11 (D)r1 (t) L21 (D)r2 (t)

(D + 1)x(t) y(t) = etx(t) + (D + 1)y(t) = 0

(D + 1)1L2 (D) = 1(D + 1)

= D2 + 2D + 2

x(t) t e1(D2 + 2D + 2)x(t) = 0 (D + 1)

= (D + 1)et = 0

y(t) (D + 1) et(D2 + 2D + 2)y(t) = 10

= (D + 1)0 et = et

y(t) t 1e yss (t) = 2= etD + 2D + 2D=1

CF S = {et cos(t), et sin(t)}

x(t) = C1 et cos(t) + C2 et sin(t)y(t) = C1 et cos(t) + C2 et sin(t) et

x(0), y(0) x (0), y (0)

x(0) = 0, y(0) = 0

x (t) + x(t) y(t) = etx(t) + y (t) + y(t) = 0

t = 0 x (0) = 1 y (0) = 0 (D2 + 2D + 2)x(t) = 0

x(0) = 0, x (0) = 1

x(t) = C1 et cos(t) + C2 et sin(t)x (t) = (C1 + C2 )et cos(t) + (C1 C2 )et sin(t)

C1 = 0 ,

C1 + C2 = 1 C2 = 1

x(t) x(t) = et sin(t)

y(t) = x (t)+x(t)et = et sin(t)+et cos(t)+et cos(t)+et sin(t)et = et cos(t)et

x(0) = 0, y(0) = 0

[x (t) = et sin (t) , y (t) = et cos (t) et ]

x y

(D + 1)x(t) y(t) = sin(t)x(t) + (D + 1)y(t) = cos(t)

(D + 1)x(t) y(t) = 10x(t) + (D + 1)y(t) = et

(D + 1)x(t) + Dy(t) = 0 (D + 1)y(t) + z(t) = 10(D + 1)x(t) + Dz(t) = et

(D + 1)x(t) + Dy(t) = 0 3x(t) + 4Dy(t) + 4z(t) = 10(D + 1)y(t) + Dz(t) = e2t

(D + 2)x(t) y(t) = 10x(t) + Dy(t) = 10et

(D2 + 4D + 3)y(t) = 30 y(0) = 0, y (0) = 0 (D2 + 4D + 3)y(t) = 30et y(0) = 0, y (0) = 0 (D2 + 4D + 4)y(t) = 30 y(0) = 0, y (0) = 0 (D2 + 4D + 3)y(t) = 30et y(0) = 0, y (0) = 0 (D2 + 2D + 10)y(t) = 30 y(0) = 0, y (0) = 0

xy + 2y + xy = x2 yc = C1 x1 cos(x)+C2 x1 sin(x) y(1) = 0, y (1) = 1

x2 y 2xy 2y = x2 ln(x) x y(1) = 1, y (1) = 1

(D + 1)x(t) y(t) = sin(t)x(t) + (D + 1)y(t) = cos(t)

x(0) = 0, y(0) = 0

(D + 1)x(t) y(t) = 10x(t) + (D + 1)y(t) = et(D + 2)x(t) y(t) = 10x(t) + Dy(t) = 10et

x(0) = 0, y(0) = 0 x(0) = 0, y(0) = 0