Help needed understanding solution-forms of non-homogeneous differential equations

Question

On my lecture material on solving non-homogeneous differential equations, I have a chart that I'm supposed to consult that tells me the the form of the attempt-particular solution. For example: If the differential equation is of the form $y''+ay'+by=sin(kx),$, my particular solution takes the form $Acos(kx)+Bsin(kx)$ if the characteristic polynomial $P(ik) \neq 0$. If the $P(ik) =0$, then the particular solution takes the form $Axcos(kx)+Bxsin(kx)$.

Why is this result true in general? I can see after doing some examples that I don't find a solution with the first form if the characteristic polynomial has the complex root involving $k$, but can we show in a simple manner why this is true, and why the addition of $x$ to the form fixes the situation?

Also, there is another curiosity: If the differential equation is of the form $y''+ay'+by=e^{kx}$,the attempted particular solution involves $Ae^{kx}$ if $P(k)\neq 0$, $Axe^{kx}$ if $P(k)=0$ but $P'(k)\neq0$, and $Ax^2e^{kx}$ if $P(k)=P'(k)=0$. Again, I can see from problem examples that these attempted solution forms do work, but what is the reason behind it?

So to summarize: Can we show, preferably by using the properties of the characteristic equation, the reasons why we don't find solutions when the characteristic equation or its derivatives take on certain values?

Answer 1

The reason is uniqueness of solutions. Once you have a valid particular solution and the fundamental system you know you have all of them.

Since this is very well known theory, but long in the details, I suggest you read up on fundamental systems of solutions for linear homogeneous equations with constant coefficients and how solutions for inhomogeneous equations are built from them and particular solutions. Most ODE books or lecture notes for mathematicians contain all the details, e.g. these lecture notes (picked at random) seem to do.

Basically one takes the higher-order homogeneous ODE and rewrites it as a linear system of first-order ODEs $u'=Au$ whose solution is essentially $e^{A x}$. The characteristic polynomial of the original ODE is in fact the characteristic polynomial of $A$, which determines its structure, hence that of the space of solutions.

Given some Ansatz for a particular solution of the inhomogeneous equation, one has then an affine space of solutions to it with the structure determined by the characteristic polynomial. Uniqueness follows then from the general theory for existence and uniqueness.