solve the differential equation Y YsinxSolutionyysinx for c

Solution

y\'\'+y\'=sinx
for complementary function
(D²+D)y=0

D(D+1)=0

=> D=0,-1

y_c.f =Ae⁰+Be^-x = A+Be^-x

for particular integral

let the solution be of the format y=Msinx+Ncosx

-Msinx - Ncosx +Mcosx-Nsinx=sinx

comparing the coefficients of sinx and cosx on both sides we get;

M+N=-1 & M-n=0

=> M=N=-1/2

y_p.i =(-1/2)*(sinx+cosx)

hence final solution is

y=y_c.f +y_p.i

or y =A+Be^-x - (1/2)(sinx + cosx)