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Find the particular solution of the differential equation y"(x) - y(x) = 1+e^x.

Question:

Find the particular solution of the differential equation {eq}y"(x) - y(x) = 1+e^x {/eq}.

Particular Solution :

A particular solution is also satisfy the given Differential Equation. We apply the new method to finding the particular solution of the nonhomogeneous differential equations. To find particular solution we use following property as follows

{eq}\displaystyle \frac{1}{f(D)}e^{bt}= \frac{e^{bt}}{f(b)},\quad f(b)\neq 0\: \: \text{if}\enspace f(b)=0\: \: \text{then}\enspace \displaystyle \frac{1}{f(D)}e^{bt}= \frac{t}{f{}'(D)}e^{bt}\\ \displaystyle D\equiv \frac{\mathrm{d} }{\mathrm{d} t}\\ D^{2}\equiv \frac{\mathrm{d^{2}} }{\mathrm{d} t^{2}} {/eq}

Answer and Explanation:

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Consider the differential equation

{eq}\displaystyle y{}''(x) - y(x) = 1+e^x {/eq}

Rewrite the differential equation as follows

{eq}\displaystyle...

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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.


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