needed help with solving linear time invariant differential equations

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needed help with solving linear time invariant differential equations

Mathematics
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obtain the solution x(t) of the differential equation
|dw:1347265100080:dw|
\[\ddot x+\omega_n^2x=t\]

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Other answers:

yup thats what i tried to write
\[m^2+\omega_n^2=0\]
would it be possible to explain that ?
the characteristic equation ?
so the first thing you would do is use the laplace transform on everything right im just not sure how to figure out how to do it for the w shaped variable
omega ? \(\omega_n^2\) is a constant for constant n
im not sure how you are going to use Laplace if you dont have any initial values
|dw:1347266422040:dw|
i knew i was forgetting something
and yeah i meant omega
well if we are going to use laplace transform you can forget the characteristic equation
ok sounds good the only difficulty i face is when things like omega come up i just get confused as to how im supposed to find the laplace transform of it
its just a positive constant, if you want to set \[\omega_n^2=A,\qquad A>0\]
ok im sorta getting it so what would the laplace transform of this constant be
the laplace transform of a constant times a function is the constant times the transfer \[\mathcal L\{αf(t)\} = α\mathcal L\{f(t)\}\] the laplace transform of one is one on the the transfer parameter \[\mathcal L\{1\}(s)=\frac 1s\]

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