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 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|<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|\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 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.

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


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\leq 1\\-1&1<t\leq 2\end{cases}}\,

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


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