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**Ragnarok** Hello, I am having great difficulty with the following problem:

$\displaystyle y''+6y'+5y=g(t)$

where $\displaystyle g(t)=t$ for $\displaystyle 0<t<2$ and $\displaystyle g(t)=0$ for $\displaystyle t>2$. Initial values are $\displaystyle y(0)=1$ and $\displaystyle y'(0)=0$.

I wrote $\displaystyle g(t)$ as $\displaystyle t-t\cdot U(t-2)$ and used Laplace transforms to get

$\displaystyle (s^2+6s+5)\mathcal{L}\{y\}-s-6=\frac{1}{s}-e^{-2s} \left(\frac{1}{s^2}+\frac{2}{s} \right)$.

Originally I distributed the $\displaystyle e^{-2s}$, solved for $\displaystyle \mathcal{L}\{y\}$ and ended up with some crazy partial fractions, but I think this definitely can't be right as I'm supposed to end up with another piecewise function as my answer. Could anyone point me in the right direction? I have never done a piecewise function with functions of t as the values.