Math 331.5: Homework 1
Complete the following for each of the dierential equations in problems 1 through 4.
(i) Graph the rate function
dy
dt
as a function of y .
(ii) Use part (i) to draw a direction eld for the dierential equation.
(iii) Based on the di
Math 331.5: Homework 22
Solutions
1. Solve the initial value problem
y + y = (t )
y (0) = y (0) = 0
Taking the Laplace transform of both sides,
L[y + y ] = L[ (t )]
L[y ] + L[y ] = es
s2 L[y ] sy (0) y (0) + L[y ] = es
(s2 + 1)L[y ] = es
L[y ] =
es
s2 + 1
Math 331.5: Homework 17
Solutions
For the following problems, we use the dierential equation
my + y + ky = g (t)
where
y is the displacement of the mass from its equilibrium position
m is the mass
is the damping coecient
k is the spring constant
g (t) is
Math 331.5: Homework 20
Solutions
1. Find the inverse Laplace transform of the following functions.
(i) F (s) =
3!
(s 2)4
f (t) = e2t t3
(ii) F (s) =
1
(s + 1)3
f (t) =
(iii) F (s) =
1 t 2
et
2
s
(s 1)2 + 4
s
s1
1
=
+
2+4
2+4
(s 1)
(s 1)
(s 1)2 + 4
1
f (t
Math 331.5: Homework 20
1. Find the inverse Laplace transform of the following functions.
(i) F (s) =
3!
(s 2)4
1
(s + 1)3
s
(iii) F (s) =
(s 1)2 + 4
(ii) F (s) =
2. Use the Laplace transform to solve the following initial value problems.
(i)
y + 4y = 2 +
Math 331.5: Homework 19
Solutions
1. Compute the Laplace transform of the following functions.
(i) f (t) = sin(bt), where b is a constant.
L[sin(bt)] =
b
, s>0
s2 + b2
(ii) f (t) = t
L[t] =
1
, s>0
s2
L[t2 ] =
2
, s>0
s3
(iii) f (t) = t2
(iv) f(t) = tn ,
Math 331.5: Homework 19
1. Compute the Laplace transform of the following functions.
(i) f (t) = sin(bt), where b is a constant.
(ii) f (t) = t
(iii) f (t) = t2
(iv) f(t) = tn , where n is a positive integer.
2. Recall that
et et
et + et
sinh t =
2
2
Comp
Math 331.5: Homework 4
Solutions
For problems 1 through 4,
(i) Find the general solution of the ODE.
(ii) Use part (i) to determine the behavior of solutions as t .
(iii) Draw a direction eld for the dierential equation and several solutions. Compare your
Math 331.5: Homework 18
1. A mass weighing 4 lb stretches a spring 1.5 in. Suppose the mass is pulled down an additional
2 in. and then released. Assume that there is no damping and the mass is acted on by an external
force of 2 cos(3t). .
(i) Determine t
Math 331.5: Homework 16
Solutions
1. Find the general solution of the dierential equation
y + y = 1 + e t + e t .
Variation of Parameters
Step 1. Solve the corresponding homogeneous equation.
yh + yh = 0
The characteristic equation is
r2 + r = 0
r(r + 1)
Math 331.5: Homework 17
1. A mass weighing 2 lb. stretches a spring 6 in. Suppose the mass is pulled down an additional 3
in. and then released, and that there is no damping.
(i) Determine the position y of the mass at any time t.
(ii) Plot y versus t.
(i
Math 331.5: Homework 15
Solutions
1. Solve the initial value problem
y 3y + 2y = 1
y (0) = 0
y (0) = 0
Step 1. Solve the corresponding homogeneous ODE.
yh 3yh + 2yh = 0
The characteristic equation is
r2 3r + 2 = 0
(r 1)(r 2) = 0
r1 = 1,
r2 = 2
t
Then the
Math 331.5: Homework 21
1. Solve the initial value problem and graph the solution.
y + 3y + 2y = u1 (t)
y (0) = 0
y (0) = 1
2. Solve the initial value problem and graph the solution.
y + y = g (t)
y (0) = y (0) = 0
where
g (t) =
0t<2
t2
t
2
3. Solve the i
Math 331.5: Homework 22
1. Solve the initial value problem
y + y = (t )
y (0) = y (0) = 0
2. Solve the initial value problem
y + 3y + 2y = (t 5)
y (0) = 0
y (0) = 1
3. Solve initial value problem
y + 9y = (t 1)
y (0) = 0
y (0) = 1
4. Solve initial value p
Math 331.5: Homework 25
1. Consider the matrix ODE
1
1
x=
4
x
1
(i) Find the general solution and describe the behavior as t .
Eigenvalues: det(I A) = 0.
Let A =
1
1
4
.
1
det(I A) = det
+1
1
4
+1
= 2 + 2 + 5
2 + 2 + 5 = 0
=
2
22 4(1)(5)
2(1)
= 1 2 i
Let
WEBASSIGN HOMEWORK
DUE: SEPTEMBER 22
These are the book problems you should complete as part of assignment 5 on
WebAssign. You should submit your answers online as instructed by WebAssign.
1. Section 7.1, page 458
48. Use integration by parts to prove the
WEBASSIGN HOMEWORK
DUE: OCTOBER 29
These are the book problems you should complete as part of assignments 13
and 14 on WebAssign. You should submit your answers online as instructed by
WebAssign.
1. Section 7.8, page 517
63. We know from Example 1 that th
Math 331.5: Homework 23
Solutions
1. Consider the predator-prey system
R = 1.5R 0.5RF
F = 0.5F + RF
(i) Find all equilibrium solutions of the system.
R(t) = 0, F (t) = 0
R(t) = 1/2, F (t) = 3
(ii) Draw a phase portrait for the system.
10
7.5
5
2.5
0
0.4
0
Math 331.5: Homework 18
Solutions
1. A mass weighing 4 lb stretches a spring 1.5 in. Suppose the mass is pulled down an additional
2 in. and then released. Assume that there is no damping and the mass is acted on by an external
force of 2 cos(3t). .
(i) D
Math 331.5: Homework 21
Solutions
1. Solve the initial value problem and graph the solution.
y + 3y + 2y = u1 (t)
y (0) = 0
y (0) = 1
Taking the Laplace transform of both sides,
L[y + 3y + 2y ] = L[u1 ]
L[y ] + 3L[y ] + 2L[y ] = L[u1 ]
s2 L[y ] sy (0) y (
Math 331.5: Homework 25
1. Consider the matrix ODE
x=
1
1
4
x
1
(i) Find the general solution and describe the behavior as t .
(ii) Sketch a phase portrait and determine if the origin is a stable spiral, unstable spiral or a
center.
2. Consider the matrix
Math 331.5: Homework 24
1. Consider the linear system
x1 = 2x1 + x2
x2 = x1 2x2
(i) Rewrite the system as a matrix equation
x = Ax
where x = (x1 , x2 ).
(ii) Find the eigenvalues 1 , 2 of the matrix A.
(iii) Find eigenvectors for A corresponding to 1 and
Math 331.5: Homework 23
1. Consider the predator-prey system
R = 1.5R 0.5RF
F = 0.5F + RF
(i) Find all equilibrium solutions of the system.
(ii) Draw a phase portrait for the system.
(iii) Describe the behavior of solutions as t and interpret your answer
Math 331.5: Homework 15
1. Solve the initial value problem
y 3y + 2y = 1
y (0) = 0
y (0) = 0
2. Solve the initial value problem
y + 6y + 13y = 5
y (0) = 0
y (0) = 1
3. Find the general solution of the dierential equation
y 6y + 10y = g (t)
for
(i) g (t) =
Math 331.5: Homework 16
1. Find the general solution of the dierential equation
y + y = 1 + e t + e t .
2. Find the general solution of the inhomogeneous ODE
1
y + 4y + 4y = 2 e2t ,
t
3. Find the general solution of the dierential equation
t > 0.
y 2y + y
Math 331.5: Homework 8
Solutions
1. Consider the ODE
(2ty 2 + 2y ) dt + (2t2 y + 2t) dy = 0
(i) Verify that this equation is exact.
M (t, y ) = 2ty 2 + 2y
N (t, y ) = 2t2 y + 2t
My = 4ty + 2
Nt = 4ty + 2
Since My = Nt , the equation is exact.
(ii) Find th
Math 331.5: Homework 8
1. Consider the ODE
(2ty 2 + 2y ) dt + (2t2 y + 2t) dy = 0
(i) Verify that this equation is exact.
(ii) Find the function F (t, y ) describing solutions implicitly via F (t, y ) = C .
(iii) Find the solution that satises the initial
Math 331.5: Homework 2
Solutions
1. Find the general solution of each dierential equation and draw several representative solutions.
(i)
dy
dt
= y + 5
First divide by the right hand side.
1
dy
=1
y + 5 dt
Next integrate with respect to t.
1
dy
y + 5 dt
(1