Method of Undetermined Coefficients: Example

Find a particular solution to the equation

Solution: Let us follow these steps:

(1)

First, we notice that the conditions are satisfied to invoke the method of undetermined coefficients.

(2)

We split the equation into the following three equations:

(3)

The root of the characteristic equation are r=-1 and r=4.

(4.1)

Particular solution to Equation (1):

Since , and , then , which is not one of the roots. Then s=0.

The particular solution is given as

If we plug it into the equation (1), we get

,

which implies A = -1/2, that is,

(4.2)

Particular solution to Equation (2):

Since , and , then , which is not one of the roots. Then s=0.

The particular solution is given as

If we plug it into the equation (2), we get

,

which implies

Easy calculations give $$A = \frac{3}{17}$$, and $$B = -\frac{5}{17}$$, that is

$$y_2 = \frac{3}{17}\cos(x) - \frac{5}{17} \sin(x)\;\cdot$$

(4.3)

Particular solution to Equation (3):

Since , and , then which is one of the roots. Then s=1.

The particular solution is given as

If we plug it into the equation (3), we get

,

which implies $$A = \frac{8}{5}$$, that is

$$y_3 = \frac{8}{5} x e^{-x}\;.$$

(5)

A particular solution to the original equation is

$$y_p = -\frac{1}{2} e^{2x} + \frac{3}{17}\cos(x) - \frac{5}{17} \sin(x) + \frac{8}{5} x e^{-x}\;.$$

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