# Ordinary Differential Equations

### From Exampleproblems

The best source of knowledge for undergraduate ODEs: video lectures from MIT.

http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/VideoLectures/index.htm

## Contents |

## Introduction

solution State and prove Gronwall's lemma.

solution Describe Picard Iteration.

## First order

solution Solve

### Nonhomogeneous

solution Solve

## Second order

solution

solution

solution

solution

solution

### Nonhomogeneous

solution

solution

solution Find a particular solution of

solution A ball is thrown straight up from the ground. How high will it go?

solution Find the general solution of the equation where .

solution

## Differential Operators

solution Evaluate

solution Evaluate

solution Find the general solution of

solution Find the general solution of

Define for and

We take *

solution Find for

solution Evaluate for

solution Find in terms of for integers a

solution Determine (i.e. P is a polynomial of degree m in λ )**

solution Determine P, Q and R are polynomials (Note that P is over r and s as above in * and R is as above in **)

## Uniqueness/Existence Theorems

solution Consider the initial value problem

where is a positive integer. Find all values of for which there is a unique solution. In the latter case, justify your answer by using an appropriate theorem.

solution Consider the initial value problem . Show that there is a unique solution when , no solution when *t*_{0} = 0 and and an infinite number of solutions when *t*_{0} = *y*_{0} = 0. Explain these reults using appropriate existence and uniqueness theorems.

solution Consider the IVP . Find all solutions.

## Lipschitz Conditions

solution Show that satisfies a Lipschitz condition when *x* lies in any bounded domain *D* (i.e. | *x* | < *M* where *M* is constant), but cannot satisfy a Lipschitz condition for all *x*.

solution Show that is continuous for all *x*, but does not satisfy a Lipschitz condition in any domain *D* which contains *x* = 0.

solution For the first-order system , show that the right-hand side satisfies a Lipschitz condition in the domain for , where and are arbitrary but finite numbers. Deduce that the IVP and has a unique solution for and obtain an estimate for . By allowing and to be as large as possible, attempt to improve your estimate of .

## Linear Systems

solution Solve the system of ODE's:

solution Write the single *n*th order equation as a first-order system.

solution Find a fundamental matrix and the Wronskian for the following linear system: .

solution Find a fundamental matrix and the Wronskian for the following linear system: .

solution Find the particular solution which vanishes at and identify the Green's matrix of the system .

solution Find a fundamental matrix, characteristic multipliers and characteristic exponents for the system

solution Find a fundamental matrix and characteristic multipliers and exponents for the system

solution Find a fundamental matrix and characteristic multipliers and exponents for the system

solution For the differential equation , where for all , show that the characteristic multipliers and satisfy the relation .

solution For the following nonlinear system, locate the critical points, classify them, and sketch the orbits near each critical point. Sketch the phase plane.

solution The motion of a simple pendulum with a linear damping is governed by the equation . where .

In the phase plane for which and , find all the critical points, classify them, and sketch the orbits near each critical point. Sketch the phase plane.

## Parametric Resonance

solution For the equation show that the conditions for parametric resonance are and or .

## Lyapunov Functions

solution Consider the system of equations

Find a Lyapunov function to determine the stability of the equilibrium solution . Consider the three cases in your calculations: a) positive semidefinite, b) positive definite and c) negative definite.

## Nonlinear

solution Solve

## Power Series

### About an irregular singular point

### About a regular singular point

## Laplace Transforms

The best introduction to the Laplace transform is from an undergrad MIT ODE class, starting at lecture 19 at the link below.

http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/CourseHome/index.htm

solution Find the Laplace transform of

solution Find the Laplace transform of

solution Find the Laplace Transform of