MATH 316 F09 Quiz 2 Solutions
1
1. Use the Laplace transform to solve the following initial-value problem:
y 6y + 9y = t, y(0) = 0, y (0) = 1.
Solution Let Y = Lcfw_y. Then Lcfw_y = sY and Lcfw_y = s2 Y 1. Taking the Laplace
transform of the dierential
MATH 316 F09 Quiz 1 Solutions
1
1
4
1. Find the general solution to the dierential equation y + x y + (1 x2 )y = 0. Express
your answer in terms of power series.
Solution Multiplying the equation by x2 we obtain x2 y + xy + (x2 4)y = 0. This is
Bessels eq
MATH 316 F09 Quiz 4 Solutions
1
1. Solve the following boundary-value problem for the wave equation
2u
2u
=
, 0 < x < L, t > 0
x2
t2
u(0, t) = 0, u(L, t) = 0
1
u
= 0.
u(x, 0) =
x(L x),
4
t t=0
a2
T
Solution Let u(x, t) = X(x)T (t). Then we have a2 X T =
MATH 316 F09 Test 1 Solutions
1
1. Verify that x = 0 is ordinary point of the dierential equation (1 + x)y + y = 0. Then
nd two linear independent power series solutions. For each solution, nd the rst four
terms of the series.
1
1
Solution In standard for
Review Problems for Final Exam
Regular Singular Points and Method of Frobenius
1. Determine whether x = 0 is a regular singular point of the differential equation x y
3
+ 2x2 y + y =
0.
Ans. No.
2. Find the general solution near x
Ans. y
= 0 of 8x2 y + 10
Review Problems for Test 2
System of Equations
1. Solve the system
dx
dt
=
Ans. x
t
2
16
t
+ 8 + C1 ,
y
=
dy
x+
t
2
dt
+ y = 0,
2
dx
dt
+ 2x + 2
dy
dt
2y = t .
t
8 + C1 .
16
2. Solve each of the following systems using Laplace transforms:
a)
y = t , x