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QUESTION

Find a particular integral of the differential equation
$\ds\frac{d^2x}{dt^2}+\frac{dx}{dt}+x=e^{2t}$.

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ANSWER

For the particular integral try $\ds x=Ce^{2t},\
\frac{dx}{dt}=2Ce^{2t},\ \frac{d^2x}{dt^2}=4Ce^{2t}$

Substituting this into the differential equation gives

$\ds 4Ce^{2t}+2Ce^{2t}+Ce^{2t}=e^{2t},\ 7C=1,\ C=\frac{1}{7}$\\
Therefore the particular integral is $\ds x=\frac{1}{7}e^{2t}$.




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