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A.2 Supplementary Exercises 345
5. Solve the IVP
u ′ +3u = δ 2 (t)+h 4 (t), u(0) = 1.
6. Find the inverse transformation of
X(s) =
s
(s 2 − 10)(s − 5)
using convolution. Write the appropriate convolution integral, but do not
calculate it.
7. Solve the initial value problem using Laplace transforms:
x ′ +2x = e −t h 3 (t), x(0) = 0.
Chapter 4 Exercises
1. Classify the type and stability of the equilibrium of the system
x ′ = −2x + y,
y ′ = −2x.
In a phase plane, draw in the nullclines (as dashed lines) and indicate which
is which. Then, noting the direction field along the x axis, sketch in a couple
of sample orbits.
2. Consider the two-dimensional linear system
( ) 1 12
x ′ =
x.
3 1
a) Find the eigenvalues and corresponding eigenvectors and identify the
type of equilibrium at the origin.
b) Write the general solution.
c) Draw a rough phase plane diagram, being sure to indicate the directions
of the orbits.
3. Use eigenvalue methods to find the general solution of the linear system
( ) 2 0
x ′ =
x.
−1 2
4. Find the equation of the orbits in the xy plane for the system x ′ =
4y, y ′ =2x − 2.