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ODE not being updated Cheat Sheet (DRAFT) by

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)