# Use variation of parameters method to find a particular solution for the differential equation ...

## Question:

Use variation of parameters method to find a particular solution for the differential equation

{eq}y'' - 9y = \frac{9x}{e^{3x}}. {/eq}

## Variation Of Parameter Method:

Consider the differential equation {eq}\displaystyle y"+q(t)y'+r(t)y=g(t) {/eq}

assume that {eq}\displaystyle y_1(t) ,y_2(t) {/eq} are a fundamental set of solutions for {eq}\displaystyle y"+q(t)y'+r(t)y=0 {/eq}

Then a particular solution to the non-homogeneous differential equation is

{eq}\displaystyle Y_p(t)=-y_1\int\frac{y_2g(t)}{W(y_1,y_2)} +y_2\int\frac{y_1g(t)}{W(y_1,y_2)} {/eq}

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Given differential equation

{eq}\displaystyle...

First-Order Linear Differential Equations

from

Chapter 16 / Lesson 3
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In this lesson you'll learn how to solve a first-order linear differential equation. We first define what such an equation is, and then we give the algorithm for solving one of that form. Specific examples follow the more general description of the method.