Solve the following constant coefficient linear differential equations using
Laplace Transform (LT), Partial Fraction Expansion (PFE), and Inverse
Laplace Transform (ILT). You must check answers in the t-domain using
the initial conditions.
For Problem 1 through Problem 5 use the initial conditions
y(0) = 3, ˙y(0) = 2.
Problem 1: (15 points)
Note: Real unique roots
y¨ (t) + 2 ˙y (t) − 8y (t) = 3
Problem 2: (15 points)
Note: Real repeated roots
y¨ (t) + 10 ˙y (t) + 25y (t) = 5e−5t
Problem 3: (20 points)
Note: Complex conjugate roots
y¨ (t) + 6 ˙y (t) + 13y (t) = 2
Problem 4: (20 points)
Note: Complex conjugate roots
y¨ (t) + 2y (t) = cos (3t)
Problem 5: (30 points)
Solve the following system of constant coefficient linear DE using LT, PFE,
and ILT.
(
5 ˙x (t) + 5y (t) = 3t
4y (t) + 2x (t) = e−2t
with the additional initial condition x(0) = 3.
MAE 460 Homework 1 Summer 2020
Practice:
Additional practice problems are provided below. Do not submit these;
they will not be graded. These are for your practice only. You can check
your work be testing the initial conditions.
Use the initial conditions from Problem 1.
Note: Real unique roots
y¨ (t) + 3 ˙y (t) + 2y (t) = e−t
y¨ (t) + 11 ˙y (t) + 30y (t) = 1.5t
y¨ (t) + 1 ˙y (t) − 30y (t) = 4t
Note: Real repeated roots
y¨ (t) − 3 ˙y (t) + 2.25y (t) = te
−2t
y¨ (t) + 4 ˙y (t) + 4y (t) = e−2t
y¨ (t) + 8 ˙y (t) + 16y (t) = 4
Note: Complex conjugate roots
y¨ (t) − 4 ˙y (t) + 5y (t) = t
y¨ (t) + 4 ˙y (t) + 8y (t) = e−2t
y¨ (t) + 10y (t) = cos (2t)
Use the initial conditions from Problem 4
(
x˙ (t) + 2y (t) = 3
y˙ (t) + 2 ˙x (t) = t
(
2 ˙x (t) − 3y (t) = t
2
y˙ (t) − 5x (t) = 1