Let denote the vector space of twice differentiable functions on . Define a linear map on by the formula

Suppose that is a basis for the solution space of . Find a basis for the solution space of the fourth order differential equation . What can you say about the kernals of and ?

This is my working out so far:

Since is a basis for the solution space of , then is a basis for the kernal of .

Therefore we can write

So since is spanned by two linearly independent vectors.

Now, consider . Since every solution of can be represented as ,

the equation becomes

, which then becomes

.

Let denote the homogenous solution and the particular solution respectively.

Obviously as is a basis for the solution space of .

Now this is where I got stuck. I did’t know what the particular solution is. I let

and tried to solve for the constants . It was a mess and I don’t think I was on the right track.

I was thinking that if I can find , then would be the basis of the fourth order differential equation . Am I correct?

Any help will be appreciated.

Thanks.