# Find a particular solution of the following differential equation: y''' - 4y' = t + 3 t + e^2t

## Question:

Find a particular solution of the following differential equation:{eq}\displaystyle \ y''' - 4y' = t + 3 \cos t + e^{2t} {/eq}

## Solving Nonhomogeneous Equations Using Undetermined Coefficients:

When looking for a particular solution to a nonhomogeneous differential equation, we need to solve the homogeneous problem first. If any terms in the form of our particular solution happen to appear in our homogeneous differential equation, those terms need to be multiplied by the independent variable before proceeding with finding the particular solution.

## Answer and Explanation:

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For the problem

{eq}\displaystyle \ y''' - 4y' = t + 3 \cos t + e^{2t} {/eq}

notice first that the solution to the homogeneous problem is given by

...

See full answer below.

#### Learn more about this topic: 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.