Math 316
Dierential Equations
Spring 2015
Homework 2 , due: Wednesday, January 21
1. Recall that that for the initial value problem y = f (x, y), y(x0 ) = y0 , we know that the
recursion
x
f (s, yn1 (s) ds
yn (x) = y0 +
x0
converges as n to the solution.
Math 316
Dierential Equations
Spring 2015
Homework 10 , due: Friday, April 3
1. Recall that a matrix N is nilpotent if N k = 0 for some k. Show that all the eigenvalues of a
nilpotent matrix are zero. Conversely, show that if all of the eigenvalues of a m
Math 316- Midterm Practice Problems
1. Find the general solution to
(tan2 x)y (2 tan x)y + (2 + tan2 x)y = 0.
2. Solve
x3 y xy + y = 0.
Solve y = 1 y 2 .
Solve y + y = sin x.
Solve y y = ex + ex .
Find the Laplace transforms of the following functions: (a
Math 316
Dierential Equations
Spring 2015
Homework 12 , due: Friday, April 17
1. Find Lyapunov functions for the following systems and use them to verify the stability of
the origin:
(i)
dx
dt
dy
dt
= x x3 y 3
,
= y + x4
(ii)
dx
dt
dy
dt
= y x cos(x2 + y
Math 316
Dierential Equations
Spring 2015
Homework 9 , due: Friday, March 27
1. Given an example of two matrices A and B for which exp(A + B) = exp(A) exp(B).
2. Show that for a projection matrix P , i.e. one for which P 2 = P , we have exp(P ) =
Id + (e
Math 316
Dierential Equations
Spring 2015
Homework 11 , due: Friday, April 10
1. Find all equilibria, classify them, and draw a local picture of the trajectories of the nonlinear
system near these points for the following systems:
dx
dt
dy
dt
(i)
= 2x + y
Math 316
Dierential Equations
Spring 2015
Homework 3 , due: Friday, February 21
1. Find the Laplace transforms of the following functions: a. f (t) = t cos t, b. f (t) = cos2 t
c. f (t) = cos3 t. d. f (t) = (t)1/2 cos(2 t). e. f (t) = (t)1/2 exp(1/4t) [Hi
Math 316
Dierential Equations
Spring 2015
Homework 7 , due: Friday, March 20
1. Solve the dierential equation
(b) A =
3 2
3 8
, (c) A =
d
x(t)
dt
5 3
3 5
= Ax(t) for the following matrices A: (a) A =
5 3
4 3
, (d) A =
d
Solve the dierential equation dt x(
Math 316
Dierential Equations
Spring 2015
Homework 7 , due: Friday, March 13
1. Find the inverse Laplace transform of the following functions using the Bromwitch Theorem:
a.
f (s) =
b.
f (s) =
s2
a
.
+ a2
s
.
(s 1)(s 2)
c.
f (s) =
s2
d.
s1
.
s2
f (s) =
2n
Math 316
Dierential Equations
Spring 2015
Homework 5 , due: Friday, February 13
1. Solve the dierential equation (1 x2 )y 6xy 4y = 0 by using the power series method.
After writing the power series representation of the two linearly independent solutions,
Math 316- Final Practice Problems
1. First Order ODEs
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Find a dierential equation of the rst order that has a general solution x2 +y 2 +cx = 0.
Solve cos2 y sin x dx + x1 dy = 0.
Solve xy + y = 2 xy.
Solve
Math 316
Dierential Equations
Spring 2015
Homework 4 , due: Wednesday, February 4
1. Solve the following dierential equations: a.(D2 +1)y = sec3 x, b. (D2 +2D+1)y = (ex 1)2 ,
c. (D2 3D + 2)y = (1 + e2x )1/2 , d. (D2 + 1)y = sec x tan x, e. (D2 3D + 2)y =
Math 316
Dierential Equations
Spring 2015
Homework 3 , due: Wednesday, January 28
1. Find the Wronskian of the functions, 1, x, x2 , x3 , . . . , xn for n > 1.
2. Which of the following sets of functions are linearly independent and which are linearly dep
Math 316
Dierential Equations
Spring 2015
Homework 1 , due: Wednesday, January 14
1. Solve the following dierential equations:
a. sin x cos y dx + cos x sin y dy = 0, b. (y x)2 dy = dx, c. (1 + x2 )y + y = tan1 x
y
y
y
d. x sin ( x ) y cos ( x ) dx + y co
Math 316- Practice Midterm
1. Solve the following dierential equations:
a. (x2 + 1) dx + x2 y 2 dy = 0
b. (sin y y sin x) dx + (cos x + x cos y) dy = 0.
c. dy dx = x2 dy + y 2 dx.
1
2
2. Solve the following dierential equations:
a. (D2 1)2 y = 0
b. (D2 4D