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APMA 0350 − HOMEWORK 3
Problem 1: (2 points) Find the general solution of an ODE whose auxiliary equation is
5r 2 (r + 4)3 (r + 7) r 2 + 93 r 2 + 2r + 102 = 0
Problem 2: (5 points) Find the eigenvalues and eigenfunctions of
y ′′ =λy
y ′ (0) =0
y(3) =0
Note: I recommend reviewing the “Mixed Example” in the lecture notes, although this is slightly different. Remember that ω > 0. That should help you figure out if you start with m = 0 or m = 1.
Problem 3: (5 points) Use undetermined coefficients to solve
y ′′ − 5y ′ + 4y =20 cos(2t) + 30 sin(2t)
y(0) = 1
y ′ (0) = 3
Problem 4: (4 points, 1 point each)
Guess the form of the particular solution (see the section “Who’s that Particular Solution?” in the lecture notes)
(a) y ′′ − 3y ′ + 2y = e t
(b) y ′′ − 3y ′ + 2y = t 2 e 2t (TURN PAGE)
(c) y ′′ − 2y ′ + 5y = sin(2t)
(d) y ′′ − 2y ′ + 5y = e t cos(2t)
Problem 5: (4 = 2 + 2 points, Mini Theory) Use the differential operators method (the one with Dy = y ′ ) to solve the following ODE
(a) y ′′ − 4y ′ + 4y = e t
(b) y ′′ − 5y ′ + 6y = e 3t
Notice how there is no guesswork involved here, which is what makes this method so nice ,
Note: Do this directly, do NOT solve the homogeneous equation
Note: Here is a video solving part (b)
Video: Cool Inhomogeneous Equations