# 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 $\displaystyle u' + 3u=0,\,\,u(0)=2\,$

### Nonhomogeneous

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

## Second order

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

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

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

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

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

### Nonhomogeneous

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

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

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

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

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

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

## Differential Operators

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

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

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

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

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

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

solution Find $\displaystyle Dy^m \,$ for $\displaystyle \, \, D = \lambda_2 +\lambda_1 + \lambda_0$

solution Evaluate $\displaystyle D \sin x \,$ for $\displaystyle \, \, D = \lambda_3 + \lambda_2^2 +\lambda_1 + \lambda_1^2$

solution Find $\displaystyle De^x, \,\,\, D= 5\lambda_4^4 + 3\lambda_2^2 + \lambda_0 \,\,\,$ in terms of $\displaystyle e^{ax} \,$ for integers a

solution Determine $\displaystyle Dx^n, \,\,\, D= P_m(\lambda_j^k) = a_m\left (\sum_{i=0}^{m} \lambda_{ij}^{mk} \right ) + a_{m-1}\left (\sum_{i=0}^{m} \lambda_{ij}^{(m-1)k} \right ) + \cdots + a_0\left (\sum_{i=0}^{m} \lambda_{ij}^{0} \right ) \,\,\,\,\, {}$ (i.e. P is a polynomial of degree m in $\displaystyle \lambda$ )**

solution Determine $\displaystyle P_n(D)Q_m(y) , \,\,\, D= R_l(\lambda_k^j) \,\,.$ 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

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

where $\displaystyle \,q$ is a positive integer. Find all values of $\displaystyle \,q$ 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 $\displaystyle ty'-t=t^2\sin t y(t_0)=y_0\,$ . Show that there is a unique solution when $\displaystyle t_0 \ne 0$ , no solution when $\displaystyle t_0=0$ and $\displaystyle y_0\ne 0$ and an infinite number of solutions when $\displaystyle t_0=y_0=0$ . Explain these reults using appropriate existence and uniqueness theorems.

solution Consider the IVP $\displaystyle ty'+y=2t, y(0)=y_0\,$ . Find all solutions.

## Lipschitz Conditions

solution Show that $\displaystyle \begin{bmatrix} 1+x_1 \\ x_2^2 \\ \end{bmatrix}$ satisfies a Lipschitz condition when $\displaystyle x$ lies in any bounded domain $\displaystyle D$ (i.e. $\displaystyle |x| where $\displaystyle M$ is constant), but cannot satisfy a Lipschitz condition for all $\displaystyle x$ .

solution Show that $\displaystyle \begin{bmatrix} \sqrt{|x_1|} \\ x_2 \\ \end{bmatrix}$ is continuous for all $\displaystyle x$ , but does not satisfy a Lipschitz condition in any domain $\displaystyle D$ which contains $\displaystyle x=0$ .

solution For the first-order system $\displaystyle x_1'=x_2, x_2'=-\sin x_1 - x_2 |x_2| + \cos t\,$ , show that the right-hand side satisfies a Lipschitz condition in the domain $\displaystyle |x|\le \beta\,$ for $\displaystyle |t|\le \alpha\,$ , where $\displaystyle \alpha\,$ and $\displaystyle \beta\,$ are arbitrary but finite numbers. Deduce that the IVP $\displaystyle x_1(0)=0\,$ and $\displaystyle x_2(0)=0\,$ has a unique solution for $\displaystyle |t|\le\delta\,$ and obtain an estimate for $\displaystyle \delta\,$ . By allowing $\displaystyle \alpha\,$ and $\displaystyle \beta\,$ to be as large as possible, attempt to improve your estimate of $\displaystyle \delta\,$ .

## Linear Systems

solution Solve the system of ODE's: $\displaystyle \begin{cases} x_t = 6x-3y\\y_t = 2x+y \end{cases}\,$

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

solution Find a fundamental matrix and the Wronskian for the following linear system: $\displaystyle \begin{cases}x_1'=x_2 \\ x_2'=x_1\end{cases}\,$ .

solution Find a fundamental matrix and the Wronskian for the following linear system: $\displaystyle \begin{cases}x_1'=(\sin t) x_2 \\ x_2'=-x_2\end{cases}\,$ .

solution Find the particular solution which vanishes at $\displaystyle t=0\,$ and identify the Green's matrix $\displaystyle G_0(t,s)\,$ of the system $\displaystyle \begin{cases}x_1'=x_2+g_1(t) \\ x_2'=-x_1+g_2(t)\end{cases}\,$ .

solution Find a fundamental matrix, characteristic multipliers and characteristic exponents for the system $\displaystyle \begin{cases}x_1'=-x_1+x_2 \\ x_2'=\left(1+\cos t - \frac{\sin t}{2+\cos t}\right)x_2\end{cases}\,$

solution Find a fundamental matrix and characteristic multipliers and exponents for the system $\displaystyle \begin{cases}x_1'=(1+2\cos 2t) x_1 + (1-2\sin 2t)x_2 \\ x_2' = -(1+2\sin 2t)x_1 + (1-2\cos 2t)x_2\end{cases}\,$

solution Find a fundamental matrix and characteristic multipliers and exponents for the system $\displaystyle \begin{cases}x_1'=\left(1+\frac{\cos t}{2+\sin t}\right)x_1 \\ x_2'=x_2+2x_1\end{cases}\,$

solution For the differential equation $\displaystyle u''+a_1(t)u'+a_2(t)u=0\,$ , where $\displaystyle a_i(t+T)=a_i(t)\,$ for all $\displaystyle t (i=1,2)\,$ , show that the characteristic multipliers $\displaystyle \rho_1\,$ and $\displaystyle \rho_2\,$ satisfy the relation $\displaystyle \rho_1\rho_2 = \exp\left\{-\int_0^T a_1(t) dt \right\}\,$ .

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

$\displaystyle x'=-2x+y^2\,$

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

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

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

## Lyapunov Functions

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

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

## Nonlinear

solution $\displaystyle y'+xy=xy^2\,$

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

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

## Power Series

### About an irregular singular point

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

### About a regular singular point

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

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

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

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

solution Find the Laplace transform of $\displaystyle f(t) = \begin{cases} 1 & 0 < t \le 1 \\ -1 & 1 < t \le 2 \end{cases}\,$

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