Problems with Trial Solutions of Nonhomogenous Differential Equations

Evaluating nonhomogenous differential equations using methods such as method of undetermined coefficients involves proposing a trial function $y\_p$ as possible solution. Here we troubleshoot and improve such trial functions that have failed in their initial test.

A Term is Already the Solution to its Homogenous Counterpart

When proposing trial solutions one often tries to choose a structure similar to the factor $G(x)$. For example solving $y^{\prime \prime }-3y^{\prime }+2y=5e^x$ we could propose $y\_p=A{e}^x$.

Troubles can arise when the chosen $y\_p$ contains a term that itself is already a solution to the homogenous counterpart of the unhomogenous differential equation. In such case the linear differential operator (the left hand side of the equation) $L\text{[}y\_p]=0$, which fails the system.

When $y\_h$ is the solution to the homogenous counterpart, the following behavior can be observed based on what constellation $y\_p$ is:

Different $y\_p$ and Solution Constellations

• $y\_p=y\_h\to$ fails
• $y\_p=y\_h+y\_other\to$ partly fails; $y\_h$-part is useless and may prevent solving uniquely
• $y\_p=y\_h\text{*}g(x)\to$ Viable if $g(x)$ changes the functions linear independence. Fails if $g(x)$ is a constant.

Standard Troubleshooting Methods

Let the root corresponding to $f(x)=L\text{[}y\_p]$ be $r\_0$. If it is the root of the characteristic polynomial with multiplicity $m$ multiply your naive trail by $x^m$.

Forcing term $f(x)$	Naive trail	If root $r\_0$ of multiplicity $m$, use
$e^{r\_0x}$	$A{e}^{r\_0x}$	$A{x}^m{e}^{r\_0x}$
$P\_k(x)e^{r\_0}x$ (poly of degree $k$)	poly of degree $k\times e^{r\_0x}$	multiply by $x^m$
$e^{r\_0x}cos(wx)\text{or}sin(wx)$	$e^{r\_0x}(Acoswx+Bsinwx)$	multiply by $x^m$

When $f(x)$ is, for example, a pure polynomial like $x^2+3x+1$ you can think of it as$(x^2+3x+1)e^{0x}$.

• When constructing trial functions, we do not use the same constant ($A,B,\text{etc.}$) twice at terms of the trial function. Do not do $y\_p=Af(x)+Ag(x)+\text{ }...$