# Ordinary Differential Equations

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

## Introduction

solution State and prove Gronwall's lemma.

solution Describe Picard Iteration.

## First order

solution Solve $u'+3u=0,\,\,u(0)=2\,$

### Nonhomogeneous

solution Solve $y'={\frac {y+x}{x}},\,\,\,y(1)=7\,$

## Second order

solution $y''+3y'+y=0\,$

solution $2y''+y'+4y=0\,$

solution $y''+4y'+3y=0,\,\,\,y(0)=1,\,y'(0)=0\,$

solution $y''+4y'+5y=0,\,\,\,y(0)=1,\,y'(0)=0\,$

solution $x^{2}y''+7xy'+8y=0\,$

### Nonhomogeneous

solution $y''+2y'+5y=e^{{-x}}\sin(2x)\,$

solution $y''-3y'+2y=3e^{{x}}\,$

solution Find a particular solution of $y''-y'+2y=10e^{{-x}}\sin(x)\,$

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

solution Find the general solution of the equation $u''+\omega ^{2}u=f(t)\,$ where $\omega \in {\mathbb {R}}\,$.

solution $xy''-y'=3x^{2}\,$

## Differential Operators

solution Evaluate $(D-1)^{3}y=0,\,\,\,D={\frac {d}{dx}}\,$

solution Evaluate $(D^{2}+5D-1)(\tan(2x)-3/x),\,\,\,D={\frac {d}{dx}}\,$

solution Find the general solution of $(2D^{2}+5D-12)y=0\,$

solution Find the general solution of $(D^{2}-1)y=2x+e^{{2x}}\,$

Define $\lambda _{k}^{j}=\left({\frac {d^{k}}{dx^{k}}}\right)^{j}$ for $\,\,k\in {\mathbb N}\cup \{0\}$ and $\,\,j\in {\mathbb Z}$

We take ${}^{r}\!\left(\lambda _{k}^{j}\right)^{s}=\lambda _{{sk}}^{{rj}},\,\,\,\,r\in {\mathbb Z},\,\,\,s\in {\mathbb N}\cup \{0\}$ *

solution Find $Dy^{m}\,$ for $\,\,D=\lambda _{2}+\lambda _{1}+\lambda _{0}$

solution Evaluate $D\sin x\,$ for $\,\,D=\lambda _{3}+\lambda _{2}^{2}+\lambda _{1}+\lambda _{1}^{2}$

solution Find $De^{x},\,\,\,D=5\lambda _{4}^{4}+3\lambda _{2}^{2}+\lambda _{0}\,\,\,$ in terms of $e^{{ax}}\,$ for integers a

solution Determine $Dx^{n},\,\,\,D=P_{m}(\lambda _{j}^{k})=a_{m}\left(\sum _{{i=0}}^{{m}}\lambda _{{ij}}^{{mk}}\right)+a_{{m-1}}\left(\sum _{{i=0}}^{{m}}\lambda _{{ij}}^{{(m-1)k}}\right)+\cdots +a_{0}\left(\sum _{{i=0}}^{{m}}\lambda _{{ij}}^{{0}}\right)\,\,\,\,\,{}$ (i.e. P is a polynomial of degree m in $\lambda$ )**

solution Determine $P_{n}(D)Q_{m}(y),\,\,\,D=R_{l}(\lambda _{k}^{j})\,\,.$ 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

${\dot {x}}=|x|^{{2/q}},x(0)=0\,$

where $\,q$ is a positive integer. Find all values of $\,q$ 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 $ty'-t=t^{2}\sin ty(t_{0})=y_{0}\,$. Show that there is a unique solution when $t_{0}\neq 0$, no solution when $t_{0}=0$ and $y_{0}\neq 0$ 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 $ty'+y=2t,y(0)=y_{0}\,$. Find all solutions.

## Lipschitz Conditions

solution Show that ${\begin{bmatrix}1+x_{1}\\x_{2}^{2}\\\end{bmatrix}}$ satisfies a Lipschitz condition when $x$ lies in any bounded domain $D$ (i.e. $|x| where $M$ is constant), but cannot satisfy a Lipschitz condition for all $x$.

solution Show that ${\begin{bmatrix}{\sqrt {|x_{1}|}}\\x_{2}\\\end{bmatrix}}$ 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 $x_{1}'=x_{2},x_{2}'=-\sin x_{1}-x_{2}|x_{2}|+\cos t\,$, show that the right-hand side satisfies a Lipschitz condition in the domain $|x|\leq \beta \,$ for $|t|\leq \alpha \,$, where $\alpha \,$ and $\beta \,$ are arbitrary but finite numbers. Deduce that the IVP $x_{1}(0)=0\,$ and $x_{2}(0)=0\,$ has a unique solution for $|t|\leq \delta \,$ and obtain an estimate for $\delta \,$. By allowing $\alpha \,$ and $\beta \,$ to be as large as possible, attempt to improve your estimate of $\delta \,$.

## Linear Systems

solution Solve the system of ODE's: ${\begin{cases}x_{t}=6x-3y\\y_{t}=2x+y\end{cases}}\,$

solution Write the single $n$th order equation $u^{{(n)}}=g(u,u',...,u^{{(n-1)}},t)\,$ as a first-order system.

solution Find a fundamental matrix and the Wronskian for the following linear system: ${\begin{cases}x_{1}'=x_{2}\\x_{2}'=x_{1}\end{cases}}\,$.

solution Find a fundamental matrix and the Wronskian for the following linear system: ${\begin{cases}x_{1}'=(\sin t)x_{2}\\x_{2}'=-x_{2}\end{cases}}\,$.

solution Find the particular solution which vanishes at $t=0\,$ and identify the Green's matrix $G_{0}(t,s)\,$ of the system ${\begin{cases}x_{1}'=x_{2}+g_{1}(t)\\x_{2}'=-x_{1}+g_{2}(t)\end{cases}}\,$.

solution Find a fundamental matrix, characteristic multipliers and characteristic exponents for the system ${\begin{cases}x_{1}'=-x_{1}+x_{2}\\x_{2}'=\left(1+\cos t-{\frac {\sin t}{2+\cos t}}\right)x_{2}\end{cases}}\,$

solution Find a fundamental matrix and characteristic multipliers and exponents for the system ${\begin{cases}x_{1}'=(1+2\cos 2t)x_{1}+(1-2\sin 2t)x_{2}\\x_{2}'=-(1+2\sin 2t)x_{1}+(1-2\cos 2t)x_{2}\end{cases}}\,$

solution Find a fundamental matrix and characteristic multipliers and exponents for the system ${\begin{cases}x_{1}'=\left(1+{\frac {\cos t}{2+\sin t}}\right)x_{1}\\x_{2}'=x_{2}+2x_{1}\end{cases}}\,$

solution For the differential equation $u''+a_{1}(t)u'+a_{2}(t)u=0\,$, where $a_{i}(t+T)=a_{i}(t)\,$ for all $t(i=1,2)\,$, show that the characteristic multipliers $\rho _{1}\,$ and $\rho _{2}\,$ satisfy the relation $\rho _{1}\rho _{2}=\exp \left\{-\int _{0}^{T}a_{1}(t)dt\right\}\,$.

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

$x'=-2x+y^{2}\,$

$y'=x-3y+y^{2}\,$

solution The motion of a simple pendulum with a linear damping is governed by the equation $u''+vu'+\omega ^{2}\sin u=0\,$. where $v>0\,$.

In the $x-y\,$ phase plane for which $x=u\,$ and $\omega y=u'\,$, 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 $u''+(\delta +\epsilon \cos t+\epsilon \sin 2t)u=0\,$ show that the conditions for parametric resonance are $\epsilon =0\,$ and $\delta =k^{2}\,$ or $(k+1/2)^{2},(k=0,1,2,...)\,$.

## Lyapunov Functions

solution Consider the system of equations ${\begin{cases}x'=y-xf(x,y)\\y'=-x-yf(x,y)\end{cases}}\,$

Find a Lyapunov function to determine the stability of the equilibrium solution $(0,0)\,$. Consider the three cases in your calculations: a) $f(x,y)\,$ positive semidefinite, b) $f(x,y)\,$ positive definite and c) $f(x,y)\,$ negative definite.

## Nonlinear

solution $y'+xy=xy^{2}\,$

solution ${\frac {1}{y}}={\frac {y''}{1+y'^{2}}}\,$

solution Solve ${\frac {dy}{dx}}=(x^{2}+y^{2})^{{1/2}}\,$

## Power Series

### About an irregular singular point

solution $x^{{3}}y''+4x^{{2}}y'+3y=0\,$

### About a regular singular point

solution $2x^{{2}}y''-xy'+\left(x^{{2}}+1\right)y=0\,$

solution $2xy''-\left(3+2x\right)y'+y=0\,$

## Laplace Transforms

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

solution $y''-y=e^{{-t}},\,\,\,y(0)=1,y'(0)=0\,$

solution Find the Laplace transform of $e^{{at}}\,$

solution Find the Laplace transform of $f(t)={\begin{cases}1&0

solution Find the Laplace Transform of $f(t)=e^{{-t}}*e^{{t}}cos(t)\,$