solving differential equations with function coefficients using Laplace Transform

Question

Does there exits a method to solve an $n$-th order liner differential equation with "function coefficients" using Laplace transform. It is well known that the identity $$L\left\{ {{t^n}f\left( t \right)} \right\} = {\left( { - 1} \right)^n}\frac{{{d^n}}}{{d{s^n}}}F\left( s \right),$$ where $L\left\{ {{t^n}f\left( t \right)} \right\}$ is the Laplace transform of $t^n f\left( t \right)$, can be used to solve such problems. However, it is not easy and indeed we didn't make any transform (we just transform D.E. with function coefficients to one another). So my Question: Does there a direct method (Like D.Es with constant coefficients) to solve this type of D.E.?

Answer 1

When all of the coefficients are either linear functions or constant functions, there is a so call "kernel method" which is modified from the Laplace transform method to solve them:

Let $y=\int_Ce^{xs}K(s)~ds$ , then you can obtain the first-order ODE for $K(s)$ . Once you find $K(s)$ , you can substitute back to $\int_Ce^{xs}K(s)~ds$ . Since the procedure in fact suitable for any complex number $s$ , you can rewrite the integral form as $\int_{a_n}^{b_n}e^{x(p_n+q_ni)t}K((p_n+q_ni)t)~d((p_n+q_ni)t)$ . By choosing suitable $x$-independent real numbers groups of $a_n$ , $b_n$ , $p_n$ and $q_n$ , you can find the linear independent solutions in integral forms.

For example in How do you solve $y''+\frac{x}{2}y'+y=0$ a 2nd order homogenous equation?.

Sometimes you need to find the linear independent solutions with the combinations of different types of integral forms , for example in Particular solution to a Riccati equation $y' = 1 + 2y + xy^2$.