Math 33B: Dierential Equations
Practice examples 15: Variation of parameters
Instructor: aliki m.
Variation of parameters is a more general method than the M.U.C for the solution of
nonhomogeneous ODEs, where the forcing term, f (x) does not need to be si

Math 33B: Dierential Equations
Practice examples 14: Reduction of order
Instructor: aliki m.
Reduction of order gives us a method to solve second order, homogeneous, linear
ODEs where the coecients of y and y may be nonconstant, provided that one of the
s

Math 33B: Dierential Equations
Review 11: Reduction of order
Feb. 12, 2014
1
Instructor: aliki m.
Introduction
The reduction of order method is a general method for solving linear, homogeneous ODEs
even when the coecients of y and y are nonconstant functi

Math 33B: Dierential Equations
Practice problems for nal
March 2014
Instructor: aliki
This handout includes 10 problems to help you prepare for the nal.
There are more Topic 1 problems as I thought Topics 2 & 3 are still fresh
in your mind. The nal will

Math 33B: Dierential Equations
Practice problems 2: Second order ODEs
Feb. 2014
Instructor: aliki m.
Problem 1
Show that ex and e2x are linearly independent solutions to,
y + y 2y = 0.
Find the particular solution that satises y(0) = 8 and y (0) = 2.
Prob

Math 33B Sample Midterm Questions
These are questions from previous years midterms. They are intended to give you an idea of
the types of questions to expect on the exam. They should not be considered as a comprehensive
study guide. Note that the exam is

Solutions to Hour Exam I
The problems on this exam came in multiple versions and were scrambled, so I
have just labelled them A, B, C,. here. The problems from your exam are here
somewhere.
A. Consider the dierential equation y = f (t, y ), where f (t, y

Math 33b Midterm solutions
1) Solve the initial value problem
y + xy cot x + x
dy
=0
dx
2
= .
2
It may be useful to consider the function u(x) = sin x.
y
Solution 1: multiply both sides of the dierential equation by sin x to get
y sin x + xy cos x + x sin

Math 33B: Dierential Equations
Practice examples 16: Simple harmonic motion
Instructor: aliki m.
Consider a spring which has been stretched 0.10 m downwards from its equilibrium
position. The object attached to it has a mass of 5 kg and the spring has a c

Math 33B: Dierential Equations
Practice examples 2: Separable ODEs
Instructor: aliki m.
Solving separable ODEs
Consider the following ordinary dierential equation,
dy
= cos2 (x) cos2 (2y).
dx
Solve the ODE using the method of separation of variables to nd

Math 33B Sample Midterm 2 Questions
These are questions from previous years midterms. They are intended to give you an idea
of the types of questions you may encounter on the exam. They should not be considered as a
comprehensive study guide. Note that th

1. (a) Find the general solution of the system y = Ay, where
1 2
4
3
A=
The characteristic polynomial is given by
2 2 + 5
The eigenvalues are = 1 + 2i and = 1 2i. The eigenvector, w, can be found from
1
(A I )w = 0. One such eigenvector is w =
. We can wr

Math 33B: Dierential Equations
Review list for Midterm 2: Second order ODEs
Feb 2014
Instructor: aliki m.
Review list
Midterm 2 will focus on solving second order ODEs. The relevant material is covered
in Lectures 10-18.
Homogeneous second order ODES:

Math 33B: Dierential Equations
Review 14: Harmonic motion
Feb 21, 2014
1
Instructor: aliki m.
Motivation
Previously, we have looked at dierent techniques to solve the second order, linear dierential equation, namely:
y + P (x)y + Q(x)y = f (x),
(1)
for si

Math 33B: Dierential Equations
Practice examples 17: Damped, unforced harmonic motion
Instructor: aliki m.
A mass of 5 kg stretches a spring 0.70 m. The mass has a damper hooked up whose
damping constant is equal to 75 kg/s. The mass is initially displace