User:Jessepfrancis/sandbox

Ordinary Differential Equations: Cheat Sheet (Using Sandbox while I write content: User/Sandbox is a real boon, this page was not there last time I was active in Wikibooks! Hope it won't get auto-deleted like main sandbox those days!)

Introduction

This book summarizes the learning on solving Ordinary Differential Equations, in much less words as possible. Those who are trying to refresh their learning in ODE might find it useful. Please feel free to update/correct information in this book, since I'm an amateur in the subject myself.

If you are looking for detailed explanation of content discussed here, please refer to the Wikibook Ordinary Differential Equations, which the author seems to have abandoned halfway, still contains a great deal of useful information in simple words. I myself learned many of the concepts given here from that book.

Also, though I have been a Administrator/Bureaucrat in one of the local language Wikibooks for a while (back in 2008) and have edited couple of books here in English Wikibooks, I'm pretty ignorant of how things work around here, and many new changes around here - please do correct me if I'm doing it wrong.

Table of Contents

1. Few Useful Definitions
   1. Wronskian of Two Functions
   2. Laplace Transforms: Definition and Properties
   3. Convolution: Definition and Properties
2. Solving First Order Ordinary Differential Equations
   1. Linear, Inhomogeneous Type
   2. Separable Equations
   3. Bernoulli's
   4. Exact Equations
3. Solving Second Order Homogeneous Ordinary Differential Equations
   1. With Constant Coefficients
   2. Euler-Cauchy Equations
4. Solving Second Order Inhomogeneous Ordinary Differential Equations
   1. Usual Method
      1. Method of Undetermined Coefficients or Guessing Method
      2. Method of Variation of Parameters
      3. Few Useful Properties for Exponential R.H.S.
   2. Using Laplace Transforms
      1. Using Convolutions
   3. Strum-Liouville Problems
      1. Green's function: Definition and Properties
      2. Solution using Green's Function
   4. Abel's identity
      1. Solving and ODE Using Abel's Identity
   5. Finding Region of Existence
      1. Picard's Theorem

Few Useful Definitions

Wronskian of Two Functions

Definition

Wronskian of two functions, $y_{1},y_{2}$ is given by $W_{y_{1},y_{2}}(x)=\left|{\begin{matrix}y_{1}&&y_{2}\\y_{1}'&&y_{2}'\end{matrix}}\right|$

Laplace Transforms

Definition

${\mathcal {L}}\{f(t)\}=F(s)=\int _{0}^{\infty }e^{-st}f(t)dt$

Properties

1. ${\mathcal {L}}\{af+bg\}=a{\mathcal {L}}\{f\}+b{\mathcal {L}}\{g\}\,,$
2. ${\mathcal {L}}\{e^{at}f(t)\}(s)=F(s-a)\,$ for $s>\alpha +a$.
3. If $F(s)={\mathcal {L}}\{f(t)\}$, then ${\mathcal {L}}\{f'(t)\}=sF(s)-f(0)$
4. Similarly, ${\mathcal {L}}\{f''(t)\}=s^{2}F(s)-sf(0)-f(0)$

Laplace Transform of Few Simple Functions

1. ${\mathcal {L}}\{1\}={1 \over s}$
2. ${\mathcal {L}}\{e^{at}\}={1 \over s-a}$
3. ${\mathcal {L}}\{\cos \omega t\}={s \over s^{2}+\omega ^{2}}$
4. ${\mathcal {L}}\{\sin \omega t\}={\omega \over s^{2}+\omega ^{2}}$
5. ${\mathcal {L}}\{1\}={1 \over s}$
6. ${\mathcal {L}}\{t^{n}\}={n! \over s^{n+1}}$

Convolution

Definition

$f(t)*g(t)=\int _{0}^{t}f(u)g(t-u)dt$

Properties

1. Associative
2. Commutative
3. Distributive over addition

Solving First Order Ordinary Differential Equations

Linear, Inhomogeneous Type

General Form

${dy \over dx}+p(x)y=q(x)$

Solution

$y(x)={\int u(x)q(x)dx+C \over u(x)}$, where

• $C$ is a constant and
• $u(x)=e^{\int p(x)dx}$

Separable

General Form

${dy \over dx}=g(x)h(y)$

Solution

Rearrange to get ${dy \over h(y)}=g(x)dx$, and integrate

Bernoulli's

General Form

${dy \over dx}+p(x)y=q(x)y^{n}$

Solution

Substitute $v=y^{1-n}$

Exact Equations

General Form

$M(x,y)dx+N(x,y)dy=0$, with ${\partial M \over \partial y}={\partial N \over \partial x}$

Solution

Solution is of the form $F(x,y)=C$, a constant, where $F_{x}=M$ and $F_{y}=N$

Solving Second Order Homogeneous Ordinary Differential Equations

With Constant Coefficients

General Form

$ay''+by'+cy=0$ or $p(D)y=0$, where

$p(D)=aD^{2}+bD+c$ is called the polynomial differential operator with constant coefficients.

Solution

1. Solve the auxiliary equation, $p(m)=0$, to get $m=\lambda _{1},\lambda _{2}$
2. If $\lambda _{1},\lambda _{2}$ are
   1. Real and distinct, then $y(x)=Ae^{\lambda _{1}x}+Be^{\lambda _{2}x}$
   2. Real and equal, then $y(x)=(Ax+B)e^{\lambda _{1}x}$
   3. Imaginary, $\lambda _{i}=a\pm bi$, then $y(x)=(A\cos {bx}+B\sin {bx})e^{ax}$

Euler-Cauchy Equations

General Form

$ax^{2}y''+bxy'+cy=0$ or $p(D)y=0$ where

$p(D)=ax^{2}D^{2}+bxD+c$ is called the polynomial differential operator.

Solution

Solving $ax^{2}y''+bxy'+cy=0$ is equivalent to solving $ay''+(b-a)y'+cy=0$