Midterm 1, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1 (10 pts.) Find the solution of the dierential equation
y = 2(1 + y )t
that satises y (0) = 1.
Solution: This equat
Quiz 8, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1 (5 pts) Find the function that has the following Laplace transform:
e15s
.
F ( s) = 2
s 14s + 74
Solution: We note th
Quiz 8, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1 (5 pts) The Laplace transform of the function
1
f (t) =
e2t et
3
t
is
s2
s 1.
Find the Laplace transform of
1
e2t et
3450:335 Ordinary Differential Equations, Kreider
You may attach additional pages if you wish. Use both sides of the paper. Label the
problems clearly and indicate your final answer/s clearly. Work al
3450:335 Ordinary Differential Equations, Kreider
You may attach additional pages if you wish. Use both sides of the paper. Label the
problems clearly and indicate your final answer/s clearly. Work al
3450:335 Ordinary Differential Equations, Kreider
You may attach additional pages if you wish. Use both sides of the paper. Label the
problems clearly and indicate your final answer/s clearly. Work al
3450:335 Ordinary Differential Equations, Kreider
You may attach additional pages if you wish. Use both sides of the paper. Label the
problems clearly and indicate your final answer/s clearly. Work al
3450:335 Ordinary Differential Equations, Kreider
You may attach additional pages if you wish. Use both sides of the paper. Label the
problems clearly and indicate your final answer/s clearly. Work al
3450:335 Ordinary Differential Equations, Kreider
You may attach additional pages if you wish. Use both sides of the paper. Label the
problems clearly and indicate your final answer/s clearly. Work al
3450:335 Ordinary Differential Equations, Kreider
You may attach additional pages if you wish. Use both sides of the paper. Label the
problems clearly and indicate your final answer/s clearly. Work al
Quiz 7, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Consider the equation
y + 2xy + 2y = 0,
and assume we can nd a solution of the form
k xk .
y (x) =
k=0
1. Find a recurrence rel
Quiz 6, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1 (5 pts). Find two linearly independent solutions of the homogeneous problem
x2 y 7xy + 41y = 0.
Solution: We assume t
Midterm 2, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1. Find the general solution of the equation
y 2y 35y = 0.
Solution: The polynomial associated to this equation is
2
Midterm 3, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1. Find the Laplace transform Y (s) = L(y )(s) of the solution y (t) the following
initial value problem:
y (0) = 4,
Solutions to Homework 1, Introduction to Dierential Equations, 3450:335-003, Dr.
Montero, Spring 2009
problem 2. The problem says that the function
y (x) = ce2x + ex
solves the ODE
y + 2y = ex ,
and a
Solutions to Homework 2, Introduction to Dierential Equations, 3450:335-003, Dr.
Montero, Spring 2009
Problem 1. The dierential equation below is exact. Find a function F(x,y) whose
level curves are s
Solutions to Homework 3, Introduction to Dierential Equations, 3450:335-003, Dr.
Montero, Spring 2009
Problem 1. Find the determinant of the matrix
3 3 9
M = 0 2 5
0
0 4
Solution: The determinant is
Solutions to Homework 4, Introduction to Dierential Equations, 3450:335-003, Dr.
Montero, Spring 2009
Problem 1. Find the solution of the equation
9y 30y + 9y = 0
with the initial data
y (0) = 1 and y
Homework 7, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1. Consider the dierential equation
(x2 + 1)y 6y = 0,
and assume there is a solution of the form
k xk .
y ( x) =
k
Solutions to Homework 8, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1. Use Laplace transform to solve the following initial value problem:
y + 5y 14y = 0, y (0) = 1, y (0
Solutions to Homework 9, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1. Given that
L
cos(5t)
t
5
e s
= ,
s
nd the Laplace transform of
t
cos(5t).
Solution: Let us call
cos
Solutions to Homework 11, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1. Solve the system
dx
=
dt
12
3 6
x,
subject to the initial condition
3
5
x(0) =
.
Solution: We comp
Quiz 1, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1 (5 pts). A bacteria population doubles its size every half an hour. If the initial
population is 3 bacteria, write do
Quiz 2, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1 (5 pts). Determine if the equation below is exact. If it is, solve it.
dy
= 0.
dx
= 2x cos(y ) and N = 2x cos(y ). Th