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

Use laplace transform to solve.

Answer 3

We're currently doing laplace to solve this stuff, so I thiiiink i figured out how to get the solution, but not the k value

Answer 4

the solution i get is: \[y(t) = \sqrt{\frac{ 16 }{ 15 }}*k*u _{1}(t)*e ^{-(t-1)/4}*\sin(\sqrt{\frac{ 15 }{ 16}}(t-1))\]

so when i plug in t = 1, y(t) = 0

Answer 5

sure.

Answer 6

so k can't = 2...

Answer 7

but there's an answer in the book

Answer 8

@agent0smith may be of more help with differential equations than I can be.

Answer 9

Thanks!

Answer 10

Wait come back @agent0smith ! lol Do I have the right idea?

Answer 11

I don't know that's why i left :P

Answer 12

dang it lol. Thanks for trying!

Answer 13

It's been a few years since I last took this course. But, solve it, like i said, and plug in ur initial conditions to solve for, \(k\).

Answer 14

alright lol. It doesn't work, so I'm just gonna write the answer down from the back of the book. But thank you for your help!