# Inverse laplace with undefined variable

I have the following information in a problem: an input $$r(t) = 60u(t)$$, with u(t) being the unit step function, and a transfer function $$H(s) = \frac{k_1}{s^{2} + s + k_{1}}$$, where k1 is a variable constant from a controller. I want to find the output y(t) but so far the farthest I can get is $$Y(s) = H(s)r(s) = \frac{k_1}{s^{2} + s + k_{1}} * \frac{60}{s}$$ Supposedly I can break this up into a partial fraction decomposition with three terms but I have no idea how to proceed because of the k1 term. Any help would be greatly appreciated

• I want to verify something a classmate of mine did, they tried going around the problem by doing $$y(t) = \mathcal{L}^{-1}(H(s)) * r(t)$$ but is that a correct method of doing it? Oct 27, 2020 at 0:00
• You can express in partial fractions as $\large \frac{A}{s}+\frac{Bs+C}{s^2 +s+k_1}$, but the form of the solution will depend on $k_1$. There are five possible forms.
– Chu
Oct 27, 2020 at 1:29
• @Chu Thank you for the reminder! :) I'm getting closer to an answer with $$Y(s) = \frac{60}{s} + \frac{-60s - 60}{s^{2} + s + k_1}$$. Thankfully the inverse laplace of the first term is just 60u(t), that just leaves the other two terms Oct 27, 2020 at 1:58

$$\dfrac{1}{s}\cdot \dfrac{k}{s^2 + s +k}\hspace{1cm} =\hspace{1cm} \dfrac{1}{s}+ \dfrac{-s-1}{s^2 + s + k}$$