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APMA 0350 − HOMEWORK 4
Problem 1: (3 points, Application)
Suppose an object of mass m = 2 is attached to a spring with constant k = 1. Moreover assume there is damping γ = 2 due to friction, and no forcing term. Assuming the initial displacement is 5 and the initial velocity is 3, solve for the displacement y(t) and describe in your own words what happens to the motion of the spring.
Problem 2: (4 points, 2 points each) Use variation of parameters to find the general solution of the following ODE. Simplify your answers.
(a) t 2 y ′′ + 4t 2 y ′ + 4t 2 y = e −2t
(b) t 2 y ′′ − t(t + 2) y ′ + (t + 2)y = 2t 3
Note: For (b) assume t and tet are solutions of the hom. equation
Problem 3: (4 points, 2 points each)
Find the general solution of y ′′ + y = cos(t)
(a) Using undetermined coefficients
(b) Using variation of parameters
Make sure your answers match
Problem 4: (3 points, Mini Theory)
In this problem, we’ll rederive the Var of Par equations, but for
y ′′ − 5y ′ + 6y = f(t)
Where f(t) is an inhomogeneous term
Recall that the hom. solution is y0 = Ae2t + Be3t
Variation of Parameters: Suppose yp is of the form
yp = u(t)e 2t + v(t)e 3t
And suppose for simplicity that
e 2tu ′ (t) + e 3t v ′ (t) = 0
Calculate (yp) ′ and (yp) ′′ and plug into the ODE to show
2e 2t u ′ (t) + 3e 3t v ′ (t) = f(t)
Problem 5: (2 points) Use tabular integration to find
Problem 6: (2 points) Use complex exponentials to find L {cos(3t)} and L {sin(3t)}
Problem 7: (2 points) Find examples of functions f(t) and g(t) with L {f(t)g(t)} ̸= L {f(t)} L {g(t)}
Most guesses should work, but try out constant/exponential functions
Problem 8: (4 points) Use Laplace Transforms to solve
y ′′ + 9y = cos(2t)
y(0) =1
y ′ (0)
Problem 9: (2 points)Find the Laplace transform of
f(t) =
t if 0 ≤ t < 2
2 if 2 ≤ t < 5
7 − t if 5 ≤ t < 7
0 if t ≥ 7
Problem 10: (2 points) Find a function whose Laplace transform is 8 /s 2 − 4s + 4
Problem 11: (2 points) Find a function whose Laplace transform is (s − 1)e −3s s 2 − 4s + 5