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QUESTION

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

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ANSWER

$\ds\frac{d^2x}{dt^2}+\frac{dx}{dt}+x=t$

To find a particular integral try $\ds x=Ct+D,\ \frac{dx}{dt}=C,\
\frac{d^2x}{dt^2}=0$

Substituting this into the ODE gives:

$0+C+Ct+D=t\Rightarrow C=1, \Rightarrow C+D=0\Rightarrow D=-1$

Hence a particular integral is $x=t-1$




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