Finding Particular Solutions to Nonhomogeneous ODEs

June 10, 2019

Undetermined Coefficients

Undetermined coefficients is also a “trial solution” like the $y=e^{rt}$ in solving $y_c$. Some examples before we head to the theories.

Examples

• Find a particular solution of $y''+3y'+4y=3x+2$

  Here the input $f(x)=3x+2$, therefore we will try $y_p=Ax+B$ same form as the input, then:

  $$y_p'=A;\ y''_p=0$$

  Substitute the trial solutions into original equation:

  $$0+3A+4Ax+4B=3x+2\\ (4A)x+(3A+4B)=3x+2$$

  Then we have $A=3/4$, $B=-1/16$, then the particular solution is

  $$y_p=\frac{3}{4}x-\frac{1}{16}$$

• Find a particular solution of $y'^{\prime}-4y=2e^{3x}$

  Input $f(x)=2e^{3x}$ then our trial solution is

  $$y_p=Ae^{3x};\ y'_p=3Ae^{3x};\ y''_p=9Ae^{3x}$$

  These give:

  $$9Ae^{3x}-4Ae^{3x}=2e^{3x}$$

  Clearly $A=\displaystyle \frac{2}{5}$ and

  $$y_p=\frac{2}{5}e^{3x}$$

• Find the particular solutions of $3y''+y'-2y=2\cos x$

  For cos and sin, we try the whole trial solution

  $$\begin{align} y_p&=A\cos x+B\sin x\\ y'_p&=-A\sin x+B\cos x\\ y''_p&=-A\cos x-B\sin x \end{align}$$

  because they appear in each other’s derivative, then

  $$\begin{align} y_p&=-\frac{5}{13}\cos x+\frac{1}{13}\sin x \end{align}$$

The above examples are designed to work. But it’s not always that lucky. If we just follow the form of $f(x)$, it may happen that the trial solution ends up in the $y_c$ that makes the equation go to zero, regardless of value of $x$. In that case we will never make $0=f(x)$. Thus we have some rules to handle these situations.

General Approach

In principle, the method of undetermined coefficients applies whenever the input $f(x)$ is a linear combination of the products of the following three types:

1. A polynomial in $x$
2. An exponential function $e^{rx}$
3. $\cos kx$ or $\sin kx$

Such function is, for example:

In practice, we first find the $y_c$ that satisfy the homogeneous solution $Ly_c=0$, and we check if $f(x)$ is in $y_c$ or any derivatives of $f(x)$ is in $y_c$, If there’s no terms in $y_c$, then we can take $f(x)$ and formulae our trial solutions.

If we found there exists a duplicate, we need to multiply the trial solution by a $x^s$ to make duplicates term disappear. $s$ should be the order of polynomial in $y_c$ plus 1.

Exponential Input theorem

$p(D)e^{\alpha x}=p(\alpha)e^{\alpha x}$ $y_p=\frac{e^{\alpha x}}{p(\alpha)}$

Exponential Shift Rule

Assuming $p(\alpha)\neq 0$. What if $p(\alpha)=0$ ? We have exponential shift rule: $p(D)e^{\alpha x}$