MATH 2454A
Problem Set 4
1. Suppose that pi (x); 0 i 2, are continuous on R and let y1 ; y2 and y3 be solutions
0
00
of y 000 + p2 (x)y 00 + p1 (x)y 0 + p0 (x)y = 0 which satisfy y1 (0) = 1; y1 (0) = 2; y1 (0) = 4;
0
00
0
00
y2 (0) = 1; y2 (0) = 0; y2 (0)
MATH 2454A
Problem Set 1
For each one of the following DEs, nd the general solution. If an initial condition is given,
then solve the initial-value problem.
1. y =
x
, y(1) = 2
y
2. y = (2x 3x2 )y
3. y = x2y 3, y(0) = 2
4. (x2 + 1)
dy
= 2xey , y(0) = 0
dx
MATH 2454A
Problem Set 2
In problems 1-8, nd the general solution. If an initial condition is given, then solve the
initial-value problem.
1
1. xy 1 + x2y = 0, y(2) = 1
2
2. x + y 2 + (2xy y)y = 0, y(2) = 0
3. 2x sin(y) + 1 + x2 cos(y)y = 0, y(2) =
4.
2
e
Ordinary Dierential Equation I - MATH2454
Carleton University
Fall 2012
Tutorial #4
Exercise 1.
Consider an autonomous equation: x = f (x). We say that c is a critical point (or equilibrium point, or stationary point) of the ODE if it is a zero of f : tha
MATH 2454A
Problem Set 7
1. For each one of the following systems, nd all critical points. Show that the system
is almost linear about each critical point. For each critical point, determine its nature
and stability for both the associated linear system a
MATH 2454A
Problem Set 6
1. Let A =
1 2
4 3
.
(a) Find a fundamental matrix for x0 = Ax by computing the eigenvectors of A.
(b) Employ the fundamental matrix found in (a) to compute eAt .
(c) Find eAt by using eAt
0
2
At
@ 0
2. Find e , where A =
0
1 4
3.
MATH 2454A
Problem Set 5
1. Express the equation x000 x2 x00 + tx0 6x + et = 0 as a system x0i = fi (t; x1 ; x2 ; x3 );
1 i 3.
2. Express the linear equation 2x000 + 4x00 5tx0 + t2 x ln(t) = 0 as a linear system
3
P
x0i =
aij (t)xj + gi (t); 1 i 3, and ex
Ordinary Dierential Equation I - MATH2454
Carleton University
Fall 2012
Midterm Exam 1h05 - October 31st
Documents & Calculator are not allowed.
Exercise 1
Study the local and global existence of solution of the following equations
y = 1 + y + y 2 cos(t)