Find the general solution. If an initial condition is given, find also corresponding particular...

Question:

Find the general solution. If an initial condition is given, find also corresponding particular solution and graph or sketch it.

{eq}y' + 2y = 4 \cos 2x, \hspace{10mm} y(\frac{1}{4}\pi) = 3 {/eq}

Method of Undetermined Coefficient:

For a given problem of the form,

{eq}f(D)y = R(x) {/eq}

where {eq}f(D) {/eq} is a differential polynomial, a particular solution {eq}y_p {/eq} which satisfies the nonhomogeneous problem can be solved by the auxiliary equation {eq}g(m) = 0 {/eq} where g is a polynomial such that

{eq}g(D)R = 0 {/eq}

This then gives us the general solution

{eq}y = y_c + y_p {/eq}

where {eq}y_c {/eq} is the solution to the homogeneous version of the equation, i.e., {eq}y_c {/eq} solves the differential equation {eq}f(D)y = 0. {/eq}

Answer and Explanation:

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For the differential equation

{eq}y' + 2y = 4 \cos 2x, \hspace{10mm} y(\frac{1}{4}\pi) = 3 {/eq}

notice first that the solution to the homogeneous...

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Undetermined Coefficients: Method & Examples
Undetermined Coefficients: Method & Examples

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Chapter 10 / Lesson 15
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The method of undetermined coefficients is used to solve a class of nonhomogeneous second order differential equations. This method makes use of the characteristic equation of the corresponding homogeneous differential equation.


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