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 equation is separable. We integrate
dy
=
1+y
2t dt + C
to ob
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 down the dierential equation governing the size of the po
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 ). This means the equation
x
2x sin(y ) + 1 + x2 cos(y )
Sol
Quiz 3, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1 (5 pts). Determine if the functions et and e2t are linearly independent in (, +).
Solution: We compute the Wronskian
W (et , e2t )(t) = det
et e2t
et 2e2t
= 2e3t e3t = e3t
Quiz 4, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1 (5 pts). Find the solution of the linear equation
y 22y + 121y = 0
that also satises y (0) = 0 and y (0) = 1.
Solution: We consider the polynomial
2 22 + 121 = ( 11)2 = 0.
Quiz 5, Introduction to Dierential Equations, 3450:335-003, Spring 2009
Question 1 (5 pts). Find the general solution of the dierential equation
y + 9y + 20y = 40e9t .
Solution: We rst look at the polynomial associated to the left hand side:
2 + 9 + 20 =
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 that the solution is of the form y (x) = xm . Plugging t
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 relation for the k .
2. Determine the rst ten k .
3. Give
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 that
s2
We nd that
L 1
1
1
1
=
= L(e7t sin(5t)(s).
2 + 25
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(5t)
f (t) =
,
t
t
cos(5t).
g (t) =
It is easy to see
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 2 35 = ( 7)( + 5).
The roots are = 7 and = 5. This mea
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, y (0) = 10.
y + 10y + 29y = 0,
Solution: We apply the
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 asks for the value of the constant c for which y (4) = 7
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 solutions to the dierential equation
y
dy
x=0
dx
Solutio
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 easy to compute along the third row. We obtain
Det(M )
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 (0) = 0.
Then, do the same for the initial data
y (0)
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=0
Find a recurrence relation for the coecients k
Fin
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) = 1.
Solution: First we apply the Laplace transform t
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
g (t) =
t
Solution: First we notice that
tf (t) = t