Ordinary Differential Equations

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

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



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 )**

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 and and an infinite number of solutions when . 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 lies in any bounded domain (i.e. where is constant), but cannot satisfy a Lipschitz condition for all .


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


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 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

solution

solution Solve


Power Series

About an irregular singular point

solution

About a regular singular point

solution

solution

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

solution Find the Laplace transform of

solution Find the Laplace transform of

solution Find the Laplace Transform of


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