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QUESTION

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


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ANSWER

For the particular integral try $\ds x=At^2+Bt+C,\
\frac{dx}{dt}=2At+B,\ \frac{d^2x}{dt^2}=2A$\\ Substituting this
into the differential equation gives $2A+At^2+Bt+C=t^2\\ t^2:\
A=1,\hspace{1cm} t:\ B=0,\hspace{1cm} \textrm{const.}:\
C=-2A=-2$\\ Therefore the particular integral is $x=t^2-2$.




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