Final Exam Practice Problems
Exam is Wednesday, May 6, 3pm–5pm
1. Verify that
1 t
4
X=
e +
tet
3
−4
is a solution to the system
′
X =
2 1
X.
−1 0
2. Find the general solution of the system
4 −5
X =
X
5 −4
′
3. Find an explicit solution of the initial-value problem
x2
dy
= y − xy,
dx
y(−1) = −1
4. Use reduction of order to find the general solution to
y ′′ − 25y = 0,
given that y1 = e5x is a solution.
1
5. Sketch, by hand, two approximate solution curves to the differential equation with the
given direction field:
• one that passes through y(−2) = 1
• one that passes through y(0) = 0.
6. Solve the differential equation
16
d4 y
d2 y
+
24
+ 9y = 0
dx4
dx2
7. Find the inverse Laplace transform of
s
(s + 1)(s2 + 4)2
8. Find the general solution of
y ′′ + 4y = sin(t)U (t − 2π),
y(0) = 1,
y ′ (0) = 0
9. Find the general solution of
y ′′ + 4y ′ + 13y = δ(t − π) + δ(t − 3π),
2
y(0) = 1,
y ′ (0) = 0
10. Find the general solution to the system
dx
= 2x − 7y
dt
dy
= 5x + 10y + 4z
dt
dz
= 5y + 2z
dt
11. Solve the following initial-value problem
y ′ + (tan x)y = cos2 x,
y(0) = −1
12. Determine whether the functions
f1 (x) = 1 + x,
f2 (x) = x,
f3 (x) = x2
are linearly independent on the interval (∞, ∞).
13. Find the general solution of the following nonhomogeneous system
−3 1
3t
′
′
X =
X + −t
2 −4
e
14. Solve the following integrodifferential equation
Z t
′
y(τ )dτ,
y (t) = 1 − sin t −
y(0) = 0
0
15. Find the general solution of
3y ′′ − 6y ′ + 6y = ex sec x
16. Consider the autonomous first-order differential equation dy/dx = y −y 3 and the initial
condition y(0) = y0 . By hand, sketch the graph of a typical solution y(x) when y0 has
the given values
(a) y0 > 1
(b) 0 < y0 < 1
(c) −1 < y0 < 0
3
(d) y0 < −1
17. Suppose the vectors
1
1
X1 = −2 + t 2 ,
4
2
1
X2 = −2 ,
4
3
2
and X3 = −6 + t 4
12
4
are solutions of a system X′ = AX. Determine whether the vectors form a fundamental
set on the interval (∞, ∞).
18. Solve the initial-value problem
2 4
′
X =
X,
−1 6
4
−1
X(0) =
.
6
0
