# 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 $x^2y'' + 7xy' + 8y = 0\,$

### Nonhomogeneous

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

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

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 $\omega\isin\mathbb{R}\,$.

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 λ )**

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 $ty'-t=t^2\sin t y(t_0)=y_0\,$. Show that there is a unique solution when $t_0 \ne 0$, no solution when t0 = 0 and $y_0\ne 0$ and an infinite number of solutions when t0 = y0 = 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 | < M 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 $|x|\le \beta\,$ for $|t|\le \alpha\,$, 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 $|t|\le\delta\,$ 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 nth 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 $f(t) = \begin{cases} 1 & 0 < t \le 1 \\ -1 & 1 < t \le 2 \end{cases}\,$

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

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