1. (6 points) Compute the inverse Laplace transforms of the given functions (using the table is recommended!). (a) F(s) = 6 (s−3)4 (b) F(s) = s+2 s2+4 (c) F(s) = 4 s3 − 5 s+3
Solve the following ODE, where δ(t) is the Dirac delta function with initial conditions y(0) = 0 and y′(0) = 0. y′′ + 2y′ − 15y = δ(t)
From the lecture, we know that L{f(t)∗g(t)} ≠ L{f(t)}∗L{g(t)}. Show that this is false by providing a counter-example. In other words, find functions f(t) and g(t) such that L{f(t)∗g(t)} ≠ L{f(t)}∗L{g(t)} and demonstrate this by computing the left hand side and right hand side using those functions to show that they are indeed unequal.
4. (4 points) Find the unit impulse response function associated with the second-order differential equation y′′ + 9y
Consider a situation in which a mass of size M is hanging from a vertical spring with spring constant K. There is a damping component on the spring which has a damping constant of C. (a) Write out a second-order homogenous differential equation to represent this situation. (b) Keeping in mind that M, K, and C are all positive, find the generalized solution of the differential equation in terms of M, K, and C. (Assume that 4MK > C2) (c) What does your solution function represent in the problem's physical context?