Cheatography

# ODE not being updated Cheat Sheet (DRAFT) by katalyst

Differential equations

This is a draft cheat sheet. It is a work in progress and is not finished yet.

### First order Differ­ential Equations

 Linear a(t)y' + b(t)y = f(t) Normal form y' + p(t)y = q(t) Separable dy/dt = g(y)*h(t) Bernoulli a(t)y' + b(t)y = f(t)ym m≠ 0,1 Homoge­neous y' = g(y/t) Exact M(x,y)dx + N(x,y)dy = 0 Exact if and only if the partials My and Nx are equal Non-Exact M(x,y)dx + N(x,y)dy = 0 When My≠Nx
First order DE's and their form

### Solving first order linear

 1. Make sure its in normal form y' + p(t)y = q(t) 2. Find an integr­ating factor µ(t) = e∫p(t)dt 3. Multiply both sides of the normal form by µ(t) to get (µ(t)y)' = µ(t)q(t) 4. Integrate both sides of (µ(t)y)' = µ(t)q(t) and solve for y
Dont Forget constants of integr­ation

### Solving FO Separable DE

 1. Rewrite y' and dy/dt and separate the variable y from the variable t to get d dy/dt =g(y)h(t)   where we get... (1/g(y)) dy = h(t)dt 2. Integrate both sides to obtain

### Second and HIgher Order DE's

 2ND Order Linear a(t)y'' + b(t)y' + c(t)y = f(t) Normal form y'' + p(t)y' + q(t)y = r(t) Homoge­neous (H) a(t)y'' + b(t)y' + c(t)y = 0 Gen. Soltn. yH(t,c­1,c2) Non-Ho­mog­eneous (NH) a(t)y'' + b(t)y' + c(t)y = f(t) Gen. Soltn. yH(t,c­1,c2) + yp(t) (H) const. coeff. ay'' + by' + cy = 0 a≠0, b,c are consts. Cauchy­-Euler at2y'' + bty' + cy = 0 a≠0, b,c are consts.
yH(t) = general solution of (H)
yP(t) = particular solution of (NH)