Reference no: EM131077540
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.
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