Solve the differential equation using undetermined coefficie

Solve the differential equation using undetermined coefficients: y\" + 25 y = 9 cos x.

Solution

First we find teh complementary solution ie solution to

y\'\'+25y=0

y\'\'=-5^2y

Genreal solution to this is

y=A sin(5x)+B cos(5x)

Now in method of undetermined coefficients we make guess for a particular solution based on inhomogeneous part ie 9 cos(x)

Let, yp=A cos(x)+B sin(x)

yp\'\'=-A cos(x)-B sin(x)

Substituing gives

24A cos(x)+24 B sin(x)=9 cos(x)

Hence, B=0, A=9/24=3/8

So, yp= 3 cos(x)/8

HEnce general solution is

y= A sin(5x)+B cos(5x)+ 3 cos(x)/8