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QUESTION

Find a particular integral of the differential equation
$\ds\frac{d^2x}{dt^2}+4x=e^{-t}$.

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ANSWER


$\ds\frac{d^2x}{dt^2}+4x=e^{-t}$

To find a particular integral try $\ds x=Ce^{-t},\
\frac{dx}{dt}=-Ce^{-t},\ \frac{d^2x}{dt^2}=Ce^{-t}$\\ Substituting
this into the ODE gives:

 $Ce^{-t}+4(Ce^{-t})=5Ce^{-t}=e^{-t}$
therefore $\ds C=\frac{1}{5}$

Hence a particular integral is $\ds x=\frac{1}{5}e^{-t}$.




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