FINAL EXAM REVIEW PROBLEMS
DUE NEVER.
1. Equations of first order
Solve the following initial value problems. If no initial value is given, find the general
solution. If you feel like it, draw a direction field and plot some of the solutions.
(1) ty 0 + 2y = sin(t),
y(π/2) = 1,
(2) ty 0 + (t + 1)y = t,
t>0
y(ln(2)) = 1,
(3) y 0 (3 + 4y) = e−x − ex ,
t>0
y(0) = 1
(4) (2xy 2 + 2y) + (2x2 y + 2x)y 0 = 0
(5) t3 y 0 + 4t2 y = e−t ,
y(−1) = 0,
t<0
(6) (3x2 − 2xy + 2)dx + (6y 2 − x2 + 3)dy = 0
(7) y 0 = (ay + b)/(cy + d),
a, b, c, d constants.
(8) sin(t)y 0 + cost(t)y = et ,
y(1) = a,
(9) y 0 − y = 1 + 3 sin(t),
(10) y 0 = x(x2 + 1)/4y 3 ,
0<t<π
y(0) = 0,
√
y(0) = −1/ 2
(11) (ex sin(y) − 2y sin(x))dx + (ex cos(y) + 2 cos(x))dy = 0
2
(12) y 0 + 2ty = 2te−t
(13) y 0 = 2x/(y + x2 y),
y(0) = −2
(14) (3x2 y + 2xy + y 3 )dx + (x2 + y 2 )dy = 0
(15) y 0 = e2x + y − 1
1
2
DUE NEVER.
2. Equations of second order
Solve the following initial value problems. If no initial value is given, find the general
solution. Try to solve the constant coefficient problems two (or even three) ways; once
using the techniques developed in class for second order equations, and once by rewriting
the problem as a first order linear system (and, if you want, using the Laplace transform).
When solutions are given, they are solutions to the associated homogeneous equation.
(1) y 00 − 2y 0 − 8y = 0
(2) t2 y 00 − t(t + 2)y 0 + (t + 2)y = 0,
(3) y 00 − 6y 0 + 82y = 0,
t > 0,
y(0) = −1,
y1 (t) = t
y 0 (0) = 2
(4) 9y 00 + 9y 0 − 4y = 0
(5) xy 00 − y 0 + 4x3 y = 0
x > 0,
y1 (x) = sin(x)
(6) y 00 + 2y 0 + 5y = 4e−t cos(2t),
(7) y 00 + 4y 0 + 5y = 0,
y(0) = 1,
y(0) = 1,
y 0 (0) = 0
y 0 (0) = 0
(8) y 00 − 2y 0 + y = et /(1 + t2 )
(9) 2y 00 + y 0 − 4y = 0,
(10) y 00 + y = tan(t),
y(0) = 0,
y 0 (0) = 1
0 < t < π/2
(11) y 00 + y = 3 sin(2t) + t cos(2t)
(12) y 00 − 2y 0 − 3y = 3te2t ,
y(0) = 1,
(13) ty 00 − (1 + t)y 0 + y = t2 e2t ,
y 0 (0) = 0
t > 0, y1 (t) = (1 + t),
y2 (t) = et
3. Series solutions
Use power series techniques to solve the following second order equations by providing
a recurrence relation to determine all the coefficients of the power series near zero. When
possible, find the general term. Classify any singular points as regular or irregular.
(1) (2 + x2 )y 00 − xy 0 + 4y = 0
(2) x2 y 00 − 5xy 0 + 9y = 0
FINAL EXAM REVIEW PROBLEMS
3
(3) (1 + x2 )y 00 − 4xy 0 + 6y = 0
(4) 2x2 y 00 + 3xy 0 + (2x2 − 1)y = 0
(5) (1 − x)y 00 + xy 0 − y = 0
(6) x2 y 00 − x(3 + x)y 0 + (x + 3)y = 0
(7) x(x − 1)y 00 + 6x2 y 0 + 3y = 0
4. The Laplace Transform
Use the Laplace transform to find solutions to the following problems. Recall that ua (t)
denotes the heaviside function centered at a, and δ(t) denotes the Dirac delta function with
unit mass at t = 0. On the exam, you will be given a list of Laplace transforms, so you
may use the table in the book.
(1) y 00 + 2y 0 + 2y = h(t),
y(0) = 0, y 0 (0) = 1 6.4.2
1 : t ∈ [π, 2π)
h(t) =
0 : t ∈ [0, π) ∪ [2π, ∞)
(2) y 00 + y = u3π (t),
y(0) = 1, y 0 (0) = 0 6.4.7
(3) y 00 + y 0 + (5/4)y = t − uπ/2 (t)(t − π/2),
(4) y 00 + 2y 0 + 3y = sin(t) + δ(t − 3π),
y(0) = 0,
y(0) = 0,
(5) y 00 + y = uπ/2 (t) + 3δ(t − 3π/2) − u2π (t),
y 0 (0) = 0 6.5.5
y(0) = 0,
5. Linear systems
y 0 (0) = 0 6.4.8
y 0 (0) = 0 6.5.9