Practice problems for MATH/MTHE 232
March 12, 2014
1. Find the general solution to the following systems of differential equations.
0 0 1
x
x
(a)
=
0
4 2
y
y
0 3 −3
x
x
=
(b)
0
2 2
y
y
0 0 1
x
x
=
(c)
0
1 2
y
y
0 5 −2
x
x
=
(d)
0
y
−2 1
y
2. For each of the systems in the previous question, find the solution that has as initial conditions
x(0) = 1 and y(0) = −1.
3. For each of the following differential equations, write it in matrix form.
d2 x dx
+
+x=0
dt2
dt
d2 x
dx
(b) − 2 + 2 + 24x = 0
dt
dt
2
dx
(c) 7 2 − 3x = 0
dt
d2 x
dx
(d) 4 2 + 3
=0
dt
dt
(a)
4. For each of the previous systems, solve it by substituting in the solution x(t) = eλt .
5. Consider the second-order system of differential equations
d2 x dx
+
+ kx = 0
dt2
dt
where k ≥ 0. Classify the solutions as overdamped, critically damped, or underdamped as k
varies in the interval [0, ∞).
1
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6. For each of the following systems, classify the origin as a saddle, source, sink, spiral source,
or spiral sink.
0 x
1 1
x
(a)
=
0
1 2
y
y
0 x
0 3
x
(b)
=
0
y
−1 2
y
0 1 x
x
−2 1
(c)
=
1
0
1 2
y
y
0 x
3 −2
x
(d)
=
0
y
1 0
y
7. For each of the following systems, draw the regions in the (a, b)-plane for which the origin is
a saddle, source, sink, spiral source, or spiral sink.
0 1 a
x
x
=
(a)
0
1 2a
y
y
0 0 a
x
x
=
(b)
0
−a b
y
y
0 a b
x
x
=
(c)
0
y
b a
y
8. Find a general solution to the following differential equations.
d2 x
dx
+ 3 + 2x = e−t
2
dt
dt
2
dx
dx
+ 5 + 4x = 3e4t
(b)
2
dt
dt
2
dx
dx
(c) 2 2 + 2 + 2x = e−7t
dt
dt
2
dx
dx
(d)
+ 18 + 81x = e−9t
2
dt
dt
(a)
9. Find a particular solution for each of the previous cases with x(0) = 1, x0 (0) = 0.
10. Find the general solution to the third-order ODE
d3 x dx
−
=0
dt3
dt
11. Find the general solution to the fourth-order ODE
d4 x d2 x
− 2 − 12x = 0
dt4
dt
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12. Consider the second-order ODE
d2 x
=0
dt2
(a) Using only your brilliant intuition, guess the obvious general solution to this ODE.
(b) Now use any of the methods that we have developed so far to solve this in a more
systematic way. How do these two solutions compare? (Hint: They had better agree, or
else you have broken Math!)