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Exercises related to differential equations, including finding general solutions and Fourier series, as well as solving wave equations with boundary and initial conditions. The exercises cover topics such as linear algebra, calculus, and physics.
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Then A (~xp + ~xh) =
The general solution to A~x = ~b is then ~xp+
(b) Find the general solution to f ′′(t) + 3f ′(t) − 4 f (t) = cos(t)
(c) Find the general solution to f ′′(t) + 3f ′(t) − 4 f (t) = t
(d) Find the general solution to f ′′(t) + 3f ′(t) − 4 f (t) = et
√^1 3 − √^13 √^1 3
,^ ~u 2 =
√^1 12 √ 2 0
, ~u 3 =
√^1 26 √ 6
Is^ {~u 1 , ~u 2 , ~u 3 }^ an orthonormal basis of^ R^3? Explain.
(b) Write ~v =
(^) as a linear combination c 1 ~u 1 + c 2 ~u 2 + c 3 ~u 3 of the ~uj ’s.
~v = ~u 1 + ~u 2 + ~u 3
(c) Verify that |~v|^2 = c^21 + c^22 + c^23
(d) Let
f (x) =
− 1 −π ≤ x < 0 1 0 ≤ x ≤ π
Find a Fourier series for f (x).
(e) Using your answer from (d), evaluate both sides of Parseval’s identity.
. Let B = A^3 + 2A + I.
Find the eigenvalues and eigenvectors of B. (You can use your work from #3(a).)
(b) Denote Df = fx Show that { √^12 , cos(nx), sin(nx)} are eigenfunctions of D^2. What are the corresponding eigenvalues?
(c) Use your answer to (b) to find eigenfunctions and eigenvalues of D^4 − 3 D^2 + 1. (Notation: (D^4 − 3 D^2 + 1)f = fxxxx − 3 fxx + f .)
(d) Solve ft = fxxxx − 3 fxx + f on 0 ≤ x ≤ π, with boundary conditions f (0, t) = f (π, t) = 0 and initial condition f (x, 0) = 2 sin(x) + 7 sin(3x)
(b) Vibrating springs usually experience damping due to air resistance and other dissipation. An example of a damped wave equation is ftt + ft = fxx.
Solve this damped wave equation on 0 ≤ x ≤ π, with boundary conditions f (0, t) = f (π, t) = 0 and initial conditions f (x, 0) = sin(x), ft(x, 0) = 0.
Compare with your answer from (a).
