Question

Diff eq help please!

Answer 1

Consider the initial value problem: \[y'' + \gamma * y' + y = k*\delta(t-1), y(0) = 0, y'(0) = 0\] where k is the magnitude of an impulse at t = 1 and gamma is the damping coefficient (or resistance).
a) Let gamma = 1/2. Find the value of k for which the response has a peak value of 2; call this value k_1.
b) Repeat part (a) for gamma = 1/4
c) Determine how k_1 varies as gamma decreases. What is the value of k_1 when gamma = 0?

Answer 2

I think I've figured out how to find the solution to the IVP, but I'm not exactly sure how to find the k_1 value.

What I found: \[y(t) = \sqrt{\frac{ 16 }{ 15 }}*k *u _{1}(t) *e ^{-(t-1)/4}*\sin(\sqrt{\frac{ 15 }{ 16 }}(t-1))\]
But it doesn't work, or I'm just doing it wrong lol