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
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)
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
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
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
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
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
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
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
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
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 positio
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
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 (
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 +
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
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
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. S
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) Dra
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
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
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 spir
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
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 th
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
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 gene
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
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 )
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 +