How do you solve second order differential equations with variables as the coefficients using a Laplace transform?

Question

Up until now I have used the Laplace transform to solve all the second order differential equations I have met. However I was recently set this equation as a homework problem and have been unable to solve it.

$ x^2y'' + 5xy' +4y=0 $

I have been made aware that you can also use a Fourier transform to solve these and am just curious as to how you solve them with both. is one method better than the other for situations like this? and how do I solve this using the Laplace transform?

Answer 1

In general, Laplace transform is useful for equations where the coefficients are constants. In this case, you can use the change of variables $x = \exp(t)$ to make this into a constant-coefficient differential equation to which you can apply the Laplace transform.