Find the general solution for the equation y 2y 3y 10 sin

Find the general solution for the equation y\" + 2y\' - 3y = 10 sin(t).

Solution

y\'\'+2y\'-3y=10

Write d/dx =D

Auxiliary equation is given by

D^2+2D-3=0

D^2+3D-D-3=0

D(D+3)-1(D+3)=0

(D+3)(D-1)=0

D=1,-3

Complimentary function is given by

y(x)=c1e^x+c2e^-3x

And for particular solution

{1/(D+3)(D-1)}10e^0=10{1/(D+3)(D-1)}e^0= 10{1/(0+3)(0-1)}=

=10/(-3)

Hence general solution is given y(x)= c1e^x+c2e^-3x-10/3