Solve 4y 36y csc3x Show me how to solve this I am lostSolu

Solve 4y\" + 36y = csc3x Show me how to solve this. I am lost.

Solution

We will solve this by variation of parameters

We can rewrite the ode as

y\'\'+9y= csc(3x)/4

First we solve the associated homogeneous ode

y\'\'+9y=0

Genral solution ot this is

yc=c1 sin(3x)+c2 cos(3x)

IN variation of parameters we use yc to make a guess for particular solution by treating constants c1,c2 as functions of x

So the guess is

yp=P(x) sin(3x)+Q(x) cos(3x)

With the constraint

P\' sin(3x)+Q\' cos(3x)=0

Q\'=-P\' tan(3x)

So,

yp\'=3P cos(3x)-3Q sin(3x)

yp\'\'=-9yp+3P\' cos(3x)-3Q\' sin(3x)

So,

yp\'\'+9yp= 3P\' cos(3x)-3Q\' sin(3x)= csc(3x)/4

Using the constraint

3P\' cos(3x)+3P\' sin^2(3x)/cos(3x)=csc(3x)/4

3P\'=cot(3x)/4

Integrating gives

3P=log(sin(3x))/12

P=log(sin(3x))/36

3P\'=cot(3x)/4

P\'=cot(3x)/12

Hence,

Q\'=-1/12

Q\'=-x/12

So,

yp=log(sin(3x))sin(3x)/36-xcos(3x)/12

General solution is

y=yh+yp