Calculate the laplace transform of the function ft e-at

Question

Math 121A: Midterm 2-

1. Consider the differential equation

y'' + 11y' + 30y = 0

for the function y(t) on the range 0 ≤ t < ∞.

(a) Calculate the Laplace transform of the function f(t) = e^-at.

(b) Determine y(t) using Laplace transforms for the conditions y(0) = 0, y'(0) = 1.

(c) Determine y(t) using Laplace transforms for the conditions y(0) = 1, y'(0) = 0.

2. Consider the function

(a) Calculate the convolution g = f ∗ f. Sketch f and g.

(b) Determine the Fourier transform of f.

(c) Determine the Fourier transform of g either by direct calculation, or by making use of standard results and your answer from part (b).

3. Consider the differential equation

y'' = f(x)

on the range 0 ≤ x ≤ 1 subject to y(0) = 0 and y'(1) = 0.

(a) Calculate a Green function solution of the form

y(x) = ₀∫¹G(x, x')f(x')dx'.

(b) Explicitly calculate the solution y(x) for the case when f(x) = x and check that this solution satisfies the differential equation and the boundary conditions.

4. (a) Calculate the Fourier series of

over the range -π < x < π, where 0 ≤ a < π.

(b) By considering Parseval's theorem and a suitable choice of a, show that

_n=1∑^∞(sin⁴ n/n⁴) = π/3 - ½.