MATH 215: MIDTERM 1: February 6th 2015
Closed Book and Notes. 50 minutes. Total 50 points
PROBLEM 1: (28 Points) Solve the following initial value problems for y(t) and then
determine the interval of existence of the solution:
i) ty 0 = y 3 ,
with y(1) =
Math 215/255: Problem set 3
1. Problem 1: For each of the following two differential equations, calculate limt y(t)
(a) y 0 + y 3 = 8 with y(0) = 4
(b) y 0 + cos(y) = 0 with y(0) = 8 .
2. Problem 2: Consider the equation
y 0 = y 2 (y 2 1) f (y)
(a) Find t
Math 215/255: Problem set 6
1. Problem 1: Consider the problem of Newtons law of cooling, where y(t) satisfies the first order
equation
y 0 + y = T0 + T1 cos(t)
with T0 , T1 constants. Find the solution of the homogeneous problem, a particular solution an
Math 215/255: Problem set 2
1. Problem 1: The speed v of a skydiver in free fall who opens a parachute at time t = T is modeled
by the the initial value problem for v(t):
mv 0 = mg kp(t)v
with
v(0) = 0
where m, g and k are positive constants and
(
p(t) =
Math 215/255: Problem set 8
Problem 1: Solve the following capacitor charging problem using Laplace transforms:
(
5 if 0 t < T
00
0
y + 6y + 5y =
0 if T t
with y(0) = 0 and y 0 (0) = 1. Here T > 0 is a parameter.
Problem 2: Solve
0 if 0 t <
00
0
y + 2y +
Math 215/255: Problem set 5
1. Problem 1: (Reduction of order) Suppose that y1 (x) is a solution to
y 00 + p(x)y 0 + q(x)y = 0
We can find a second (independent) solution as follows. We will assume that y1 (x) 6= 0 for x in the
range we are considering. T
MATH 215: MIDTERM 2: MARCH 20th 2014 (R. Froese, M. Ward)
Closed Book and Notes. 50 minutes. Total 50 points
PROBLEM 1: (20 Points) Let be a positive constant, and let y(t) solve
y + 0.1 y + 4y = sin(t) ,
t 0.
i) Calculate the steady-state solution in the
Math 215/255: Problem set 4
1. Problem 1: Consider x2 y 00 x(x + 2)y 0 + (x + 2)y = 0 for x > 0.
(a) Verify that y1 = x and y2 = xex are solutions
(b) By calculating the Wronskian show that y1 and y2 form a fundamental set of solutions (that is,
they are
Math 215/255: Problem set 1
1. Problem 1: For each positive integer n, find the solution of y 0 = y n with y(0) = 1. For what values
of x is the solution defined?
2. Problem 2: Find the general solution to y 0 = xy + x + y + 1.
3. Problem 3: Solve y 0 = 2
Math 215/255: Problem set 7
Problem 1: Find the inverse Laplace transform f (t) = L1 [F (s)] where
2s 3
+ 2s + 10
1
2. F (s) = 2
9s 12s + 3
2s + 1
3. F (s) = 2
4s + 4s + 5
1. F (s) =
s2
Problem 2: Use the Laplace transform method to solve y 00 + 2y 0 + 5y
Homework 2: MATH 215/255
Due in class on Friday, January 29th
Talking to other students about the problems is encouraged but you must submit your own work and
identify who you worked with at the top of your assignment.
Compile your work into a single docu
MATH 215/255 - section 99 : Quiz 2, February 24 2016
Duration : 15 minutes. No lecture notes, no textbooks, no calculators.
Content : 2 problems, Total : 25 points
Write clearly your nal answer in the box for each question. Only the answers written in the
MA 215/255 Spring 2016
Quiz #1 - February 1, 2016
Circle or otherwise clearly indicate your answer on the page.
Use the back of the paper to show your work as needed.
ID NUMBER:
PRINT YOUR NAME:
1. (10 points) Consider the following for x > 0,
1 dy
2
y =
HOMEWORK 3: Math 215 February 2016 Solutions
1. Analytically, nd the solution y(t) to
y =
3
1
2
2
y,
with y(0) =
1
1
.
Use Matlab to plot the direction eld for this system. Overlay your analytic solution for this particular
initial condition on the direct
HOMEWORK 1 SOLUTIONS: MATH 215/255
Show all relevant work for credit. You will be marked for your work and your answer as appropriate.
Talking to other students about the problems is encouraged but you must submit your own work and identify
who you worked
Homework 2: MATH 215/255 Solutions
Talking to other students about the problems is encouraged but you must submit your own work and
identify who you worked with at the top of your assignment.
Compile your work into a single document and print it. Staple p
Math 215/255: Homework #4 Due Friday, March 4, 2016.
1. A competing species model. A stylized model of competing species with population densities
x(t) and y(t) is given by
dx
= x(1 x y),
dt
dy
1 y 3x
=y
dt
2 4
4
.
First, nd all four critical points. Th