y "(x) - 3 y '(x) + 2 y(x) = e^x sin x

Question:

{eq}y"(x)-3y'(x)+2y(x)=e^x\sin x {/eq}

Solving Non-homogeneous Equations Using Undetermined Coefficients:

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 problem itself can be obtained by solving 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 obtained by solving the auxiliary equation

{eq}f(m) = 0. {/eq}

Answer and Explanation:

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For the ODE

{eq}y"(x)-3y'(x)+2y(x)=e^x\sin x {/eq}

the auxiliary equation for the homogeneous counter part is given by

{eq}m^2 - 3m + 2 = 0...

See full answer below.


Learn more about this topic:

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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