Laplace Transform (LT), Partial Fraction Expansion (PFE), and Inverse

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