SM212, Prof Joyner, 2-27-2004
Practice Test 2
PROBLEM 1: Find the general form of yp . Do not solve for the undetermined coecients. Do not solve the ODE.
(a) y 2y + y = xex ,
(b) y 3y + 2y = ex + e2x + sin(x).
(Ans: (a) yp = A1 x3 ex + A2 x2 ex + A3 xex ,
Sm212, Review 2
George Nakos
20 September, 2002
This is a set of review problems. It may not be sucient preparation for
the test, but it covers most basic topics.
1. Give the denition of the functions f1 (x) , f2 (x) , . . . , fn (x) being linearly depend
SM212 Practice Test 3, Prof Joyner
1. Find the Laplace transform of f (t), where
(a) f (t) = t2 sin(t) (the convolution of t2 and sin(t),
(b) f (t) = sin(t)u(t 2 ).
2. Let
f (t) =
1,
0 < t < 1,
0, t 0 or t 1.
Solve x + 4x = f (t), x(0) = x (0) = 0.
1
s
3.
SM 212 Final Examination
11 December 1996
1. (a) Find the general solution to
(i) x2 dy/dx = 4 3xy ,
(ii) (1 x cos(xy )dy/dx = 1 + y cos(xy ),
(b) Solve the initial value problem dy/dx = 2x cos2 (y ), y (0) = /4.
2. (a) A tank contains 100 gallons of wate
SM212 Final Spring 2003-2004 Solutions
Written portion. Each problem is worth 20 points (Chairman says everyone must use same point
grading scheme)
1. Taking LTs of each DE gives (s 3)X (s)+6Y (s) = 1, X (s) = (s 3)Y (s), so (s 3)2 +6)Y (s) = 1,
1
ie, Y (
SM212/P Final Examination: Multiple Choice
Section
04 May 2009 1930
The Multiple Choice part of the exam counts 50%. Fill in the letter of the
best answer on the bubble sheet. There is no penalty for wrong answers.
1. Which of the following dierential equ
Initial value problems
Prof. Joyner, 8-17-20071
A 1-st order initial value problem, or IVP, is simply a 1-st order ODE
and an initial condition. For example,
x (t) + p(t)x(t) = q (t),
x(0) = x0 ,
where p(t), q (t) and x0 are given. The analog of this for
Existence of solutions to ODEs
Prof. Joyner1
[Peano] was a man I greatly admired from the moment I met
him for the rst time in 1900 at a Congress of Philosophy, which
he dominated by the exactness of his mind.
-Bertrand Russell, 1932
When do solutions to
Numerical solutions to DEs
Eulers method and improved Eulers method
Prof. Joyner1
Read Euler: he is our master in everything.
- Pierre Simon de Laplace
Leonhard Euler (pronounced Oiler) was a Swiss mathematician who
made signicant contributions to a wide
First order ODEs - separable and linear cases
Prof. Joyner, 8-21-20071
Separable DEs:
We know how to solve any ODE of the form
y = f (t),
at least in principle - just integrate both sides2 . For a more
general type of ODE, such as
y = f (t, y ),
this fail
Newtonian mechanics - falling body problems
Prof. Joyner1
We briey recall how the physics of the falling body problem leads naturally to a dierential equation (this was already mentioned in the introduction
and forms a part of Newtonian mechanics [M]). Co
Linear ODEs, I
Prof. Joyner, 8-17-20071
We want to describe the form a solution to a linear ODE can take. Before
doing this, we introduce two pieces of terminology.
Suppose f1 (t), f2 (t), . . . , fn (t) are given functions. A linear combination of these
Linear ODEs, II
Prof. Joyner, 8-18-20071
To better describe the form a solution to a linear ODE can take, we
need to better understand the nature of fundamental solutions and particular
solutions.
Recall that the general solution to
y (n) + b1 (t)y (n1) +
Applications of DEs:
Simple LRC circuits
Prof. Joyner1
An LRC circuit is a closed loop containing an inductor of L henries, a
resistor of R ohms, a capacitor of C farads, and an EMF (electro-motive
force), or battery, of E (t) volts, all connected in seri