Method of Undetermined Coefficients: Example

Find a particular solution to the equation

displaymath61

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:

displaymath63

(3)
The root of the characteristic equation tex2html_wrap_inline65 are r=-1 and r=4.
(4.1)
Particular solution to Equation (1):
Since tex2html_wrap_inline73, and tex2html_wrap_inline75 , then tex2html_wrap_inline77, which is not one of the roots. Then s=0.
The particular solution is given as

displaymath81

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

displaymath85,

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

displaymath89

(4.2)
Particular solution to Equation (2):
Since tex2html_wrap_inline93, and tex2html_wrap_inline95 , then tex2html_wrap_inline97, which is not one of the roots. Then s=0.
The particular solution is given as

displaymath101

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

displaymath105,

which implies

displaymath107

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

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

(4.3)
Particular solution to Equation (3):
Since tex2html_wrap_inline117, and tex2html_wrap_inline75 , then tex2html_wrap_inline121 which is one of the roots. Then s=1.
The particular solution is given as

displaymath125

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

displaymath129,

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

\begin{displaymath}y_3 = \frac{8}{5} x e^{-x}\;.\end{displaymath}

(5)
A particular solution to the original equation is


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

[Differential Equations] [First Order D.E.] [Second Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Mohamed Amine Khamsi

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour