MAT242, Fall 2012. Quiz # 4, SOLUTIONS
1. (12 pts) Draw a possible phase portrait for a system satisfying
all the properties listed below. Clearly indicate the equilibrium points
and the behaviour of typical trajectories.
There are only two equilibrium p

MAT 242, Fall 2011. Quiz # 5, SOLUTIONS
1. (20 pts) Use the eigenvalue method to nd the general solution of
the following systems:
a)
[
1 2
4 1
x =
.
]
x
Solution. The characteristic equation is
(1 )(1 ) 8 = 0,
2 9 = 0.
The eigenvalues are 1 = 3, 2 = 3. T

MAT 242, Fall 2011. Quiz # 4, SOLUTIONS
1. (14 pts) Find the general solution of the non-homogeneous linear
equation:
y + 3y + 2y = 2ex + cos x.
Solution. The homogeneous equation is:
.
y + 3y + 2y = 0.
The characteristic equation is
r2 + 3r + 2 = 0.
It h

MAT 242, Fall 2012. Quiz # 3, SOLUTIONS
1. (10 pts) Find the general solution of the non-homogeneous linear
equation:
y + 3y + 2y = cos x.
Solution. The homogeneous equation is:
.
y + 3y + 2y = 0.
The characteristic equation is
r2 + 3r + 2 = 0.
It has sol

Quiz # 2, SOLUTIONS
1. (12 pts) Consider the dierential equation
dy
(8y + x) + (3 + y) = 0
dx
a) Verify that the equation is exact
Solution.
.
(3 + y) = 1 =
(8y + x),
y
x
so the equation is exact.
b) Find the general solution.
Solution. For the unknown fu

MAT 242, Fall 2011. Quiz # 3, SOLUTIONS
1. (12 pts) Find the general solution of the dierential equation
y (5) + 8y (4) + 17y (3) = 0.
Solution. The characteristic equation is
r5 + 8r4 + 17r3 = 0,
.
r3 (r + 4)2 = 1,
r = 4 i and r = 0 repeated 3 times.
The

MAT 242, Fall 2012. Quiz # 2, SOLUTIONS
1. (10pts) Consider the dierential equation
dy
(9y 2 + x) + (6x + y) = 0.
dx
a) Verify that the equation is exact.
Solution. We can re-write the equation as
.
(9y 2 + x)dy + (6x + y)dx = 0.
We have
(9y 2 + x) = 1 =

Quiz # 1, Solutions
1. (10 pts) Consider the initial value problem
dy
dx
= y 3 x, y(0) = 0.
1
a) Does the Existence and Uniqueness Theorem guarantee the uniqueness of the solution? Justify your answer carefully, a yes/no answer
will not suce.
b) . Solve t

Midterm test SOLUTIONS
1. (20 pts)
a) Write a dierential equation describing a function y(x) with the
following property: the slope of the tangent to the graph at a point
(x, y) is the product of the squares of the coordinates x and y.
b) Give an example

Midterm test SOLUTIONS
1. (18 pts)
a) Write a dierential equation describing a function y(x) with the
following property: the slope of the tangent to the graph at a point
(x, y) is the product of the coordinates x and y.
Solution.
.
dy
= xy.
dx
b) Give an

Quiz # 1, Solutions
1. (8 pts) Write a dierential equation for the function y(x) which
has the following property:
the tangent line to the graph of y(x) at a point (x, y) has a slope which
is the sum of the squares of x and y.
You do not have to solve the