Solve the differential equation {eq}y''' - 3y = \cos x {/eq}

Question:

Solve the differential equation {eq}y''' - 3y = \cos x {/eq}

Solution Of The Linear Differential Equation:

The given differential equation second order linear differential equation. There are various methods of solving differential equations. To find the solution of the differential equation first we find the complementry function and particular integral then the general solution of the differential equation will be written as follows.

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

To find the particular integral we use following property

{eq}\left [ \frac{1}{f(D^2)}\sin(at)=\frac{1}{f(-a^2)}\sin(at),\: \: f(-a^2)\neq 0\enspace \text{if}\enspace f(-a^2)=0\enspace \text{then}\enspace \frac{1}{f(D^2)}\sin(at)=\frac{x}{f{}'(D)}\sin(at)\right ]\\ {/eq}

Answer and Explanation: 1

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Consider the differential equation

{eq}y''' - 3y{}' = \cos x {/eq}

Rewrite the differential equation as follows

{eq}(D^{3}-3D)y=\cos x {/eq}

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First-Order Linear Differential Equations

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Chapter 16 / Lesson 3
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Learn to define what a linear differential equation and a first-order linear equation are. Learn how to solve the linear differential equation. See examples.


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