Please show all steps solve the following diffential equatio

Please show all steps

solve the following diffential equation using Undetermined Coefficients y``+9y=sin 3x

Solution

First we find solution to the homogeneous ODE ie find the complementary solution.

y\'\'+9y=0

y\'\'=-3^2y

General solution to this ODE is:

y=A sin(3x)+B cos(3x)

Now we find the complementary solution.

IN the method of undetermined coefficients. We make a guess for particular solution based on the inhomogeneous part of the ODE. In this case it is sin(3x)

The guess usually is: C sin(3x)+D cos(3x)

But since sin(3x) is already a solution of the homegenous ODE we multiply our guess by x

So our guess becomes:

yp=Cx sin(3x)+Dx cos(3x)

yp\'=C sin(3x)+D cos(3x)+3x( Ccos(3x)- Dsin(3x))

yp\'\'=3C cos(3x)-3D sin(3x)-9x( sin(3x)+ cos(3x))+3( Ccos(3x)- Dsin(3x))

yp\'\'=3C cos(3x)-3D sin(3x)-9yp+3( Ccos(3x)- Dsin(3x))

yp\'\'+yp=3C cos(3x)-3D sin(3x)+3( Ccos(3x)- Dsin(3x))=sin(3x)

Comparing coefficients gives:

6C=0 ie C=0

-6D=1 ie D=-1/6

So, the general solution is:

y=A sin(3x)+B cos(3x)-sin(3x)/6