Finding the impulse response, given a differential function involving the input and output I'm starting to teach myself some signal processing, and I'm stuck on this problem where I have to find the impulse response $h(t)$ of a causal LTI system given: $$y''(t)+2y'(t)+y(t)=x(t)$$ So far what I've done is taken the Laplace transform of both sides of this equation to get $$(s^2+2s+1)Y(s)-3sy(0)-y'(0) = X(s)$$ So the problem now is that I have all these inital values that I don't know what to do with. If I understand correctly, for all $ t&lt;0$ in a causal system we have the output as zero, but this doesn't include zero. If anybody can point me in the right direction, I would really appreciate it. Thanks.
If you do not select
$$
y'(0) = 0\\
y(0) = 0
$$
then the system will not be a linear, time-invariant filter, so one usually makes that selection and then deals with the special cases for when the initial conditions are not zero.