differential equation solve use Greens function do not eval

differential equation: solve use Green\'s function ( do not evaluate integral defining yp )

y\'\'-16y = xe^-2x

Solution

Solution: 28/04/2016

The given equation is of the form f(D)y = 0

The auxiliary equation is

M² -16 = 0

m = -4, +4

the complementary solution n of (1)

y_c = c₁ e^4x + c₂ e^-4x

y_p = 1/(D² -16) ( xe -2x )

= 1/( D² -16 ) xe – 2/ (D² -16) x

= ex/(D+1)² – 16 )-2x /(D²-16)

= ex/ (D² +2D +1-16) -2x /(D²-16)

= ex/ (D² +2D -15) – 2x / (D² -16)

= ex/ -15 ( 1-(D²+2D) /15 ) -2x / -16(1-D²/16)

= -e/15 x ( 1-(D²+2D) /15 )^-1 + x/8 (1-D² /16 )^-1

= -e/15 [ 1+ (D²+2D/15 ) + ……. ] x +x/8 [ 1+ D²/16 + ………]

= -e/15 [ x+2/15 ] + x/8

= -e/225 ( 15x +2 ) + x/8

Therefore G.S of (1) is

Y = C₁ e^4x+ c₂e^-4x -e/225 ( 15x +2 ) + x/8