# Use variation of parameters to solve the initial value problem. y"+4y'+4y=(3+x)e^{-2x} , \;...

## Question:

Use variation of parameters to solve the initial value problem.

{eq}y"+4y'+4y=(3+x)e^{-2x} , \; y(0)=2, \; y(0)=5 {/eq}

## Solution of the Differential Equation:

The given differential equation is a second order linear differential equation. To find the solution of the differential equation, we first find the auxiliary equation and then write complementary solution in the form as follows:

{eq}y_c=(c_1+c_2x)e^{\beta x} {/eq}

To find particular solution we use following property as follows:

{eq}f=-\int \frac{vR}{W}dx\\ g=\int \frac{uR}{W}dx {/eq}where W is the Wronskian of the homogeneous solutions.

Then, the general solution of the differential equation will be in the form of

{eq}y(x)=y_c+y_p{/eq}

## Answer and Explanation:

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Consider the differential equation

{eq}y{}''+4y{}'+4y=(3+x)e^{-2x} , \; y(0)=2, \; y'(0)=5 {/eq}

Rewrite the differential equation as follows

{eq}...

See full answer below.

#### Learn more about this topic: First-Order Linear Differential Equations

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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.