MATH 325 ODE

First Order Nonlinear

First Order ODE

Nth Order ODE

Second Order ODE and IVPs

Introduction

First Order Nonlinear

Existence and Uniqueness

Linear Homogeneous Constant Coefficient

Linear Nonhomogeneous Second Order

Nonlinear Second Order

Linear Homogeneous Nth Order

Abel's Theorem

Linear Nonhomogeneous Nth Order

The Fundamental Set of solutions

Series Solutions

Ordinary Points

Nonhomogeneous Case

Review of Power Series

Regular Singular Points

Euler Equations: $x^2y''+p_0 xy' + q_0 y=0$

Frobenius' Method: multiply $(x-x_0)^2 $ on both sides

Laplace Transforms

Solving Constant Coefficient ODE

Introduction

Convolutions

Linear Systems of ODEs

Constant Coefficient Homogeneous Systems

Nonhomogeneous Constant Coefficient Systems

Introduction

Matrix Exponentials

Stability

Linearization

Initial Value Problem

N-th Order ODE and First Order System of ODEs

Homogeneous

Bernoulli

Exact (with integrating factor)

Seperable

Other substitutions (u=Ax+By+C)

Solve ODEs with Picard Iteration

y'' = f( y , y' ) doesn't depend on t

y'' = f( t , y' ) doesn't depend on y

Method of undetermined coefficient

Variation of parameters

Repeated root: $y_1=e^{r_1t}$, $y_2= te^{r_1t}$ (reduction of order)

First Order Linear

Complex roots: $y_1=e^{\alpha t}cos(\beta t)$, $y_2=e^{\alpha t}sin(\beta t)$

Distinct roots: $y_1=e^{r_1t}$, $y_2=e^{r_2t}$

$y=k_1 y_1 + k_2 y_2$

Auxiliary Equation

Complementary Solution

Find particular solution (look up table)

Only works for constant coefficient

Need to know $y_1, y_2$

Particular solution $y_p = u_1(x)y_1(x)+u_2(x)y_2(x)$

Compute $u_1,u_2$ using formula

Can solve all kinds of ODE

Span all the solutions

Linearly independent (Wronksian is not 0)

Particular Solution: $ y_p = \sum_{i=1}^{n} u_i(x) y_i(x)$

Cramer's Rule: $ u_i = \int \frac{w_i(x)}{w(x)} \,dx $

Constant Coefficient: auxiliary equation

Non-constant Coefficient: series solution

Let $ y = \sum_{n=0}^{\infty} a_n x^n$, substitute it into L[y]

Require $L[y] = y''+p(x)y'+q(x)y $: p(x), q(x) analytic

$y''+p(x)y'+q(x)y =g(x)$. Expand g(x) by power series

Indicial Equation: $F(r) = r(r-1) + rp_0+q_0=0 $

General Solution: $ |x-x_0|^r\sum_{n=0}^{\infty} a_n (x-x_0)^n $ or let $x>x_0, a_0=1$ then $ y=(x-x_0)^r[1+\sum_{n=1}^{\infty} a_n (x-x_0)^{n}] $

First/Second Translation Theorem