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
Review for the rst midterm, Introduction to Di. Eqs., 3450:335-003, Dr. Montero,
Spring 2009
Separable equations
A separable equation is an equation of the form
dy
= h(t)g(y),
dt
where h(t) depends only on t and g(y) depends only on y. We nd an implicit s
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
Review for the third midterm, Introduction to Di. Eqs., 3450:335-003, Dr. Montero,
Spring 2009
Some useful facts regarding the Laplace transform.
For a function f (t) dened on [0, ), the Laplace transform L(f ) is the function of s dened
through the integ
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 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 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)
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
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
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
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 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
Solutions to Homework 6, Introduction to Dierential Equations, 3450:335-003, Dr.
Montero, Spring 2009
Problem 1. Find a particular solution to
y + 6y + 9y =
25e3t
.
2(1 + t2 )
Solution: First we nd the solutions to the homogeneous equation
y + 6y + 9y =
Solutions to Homework 5, Introduction to Dierential Equations, 3450:335-003, Dr.
Montero, Spring 2009
Problem 1. Find a particular solution to the dierential equation
y 4y = 48t3 .
Solution: First we look at the polynomial associated to the right hand sid