How do I solve $y''+y'+7y=t$? How do I solve $y''+y'+7y=t$ where $y(0)=0$ and $y'(0)=0$ $(t\geq 0)$?
I tried to solve this by Laplace transformation, but I couldn't find the inverse of $1/(s^2(s^2+s+7))$.
How would I solve this?

The homogeneous equation shouldn't be a problem to you.
For a particular solution, notice that the derivative of the RHS is a constant, and the second derivative vanishes. So plugging $y=t$ you get
$$0+1+7t=t$$
not so far from the solution. You fix by adjusting the coefficient of $t$ to $t/7$ and adding a constant to compensate the $1$:
$$0+\frac17+7(\frac t7+c)=t,$$then
$$\frac t7-\frac1{49}.$$
This trial-and-error process (an informal application of the indeterminate coefficients method) will work with polynomials and exponentials. First try $y=RHS(t)$, observe the patterns and fix the discrepancies.
