Differential equations

First Order Linear Differential Equations:
1. Separating variables
Example 1:
$\dfrac{dy}{dx}=f(y)\\ \dfrac{dy}{f(y)}=dx\\ \int \dfrac{dy}{f(y)}=\int dx$

Example 2:
$\dfrac{dy}{dx}=\dfrac{1}{f(x)f(y)}\\ f(y)dy=\dfrac{dx}{f(x)}\\ \int f(y)dy=\int \dfrac{dx}{f(x)}$

2. Integrating factors

To solve equations of the form
$a(x)\dfrac{dy}{dx}+b(x)y=c(x)$

i. Express in standard form
$\dfrac{dy}{dx}+p(x)y=q(x)$

ii. Multiply both sides by the integrating factor $e^{\int p(x) dx}$

iii. $\dfrac{d}{dx}(ye^{\int p(x) dx})=q(x)e^{\int p(x) dx}$

iv. $ye^{\int p(x) dx}=\int q(x)e^{\int p(x) dx}+C$

v. Divide both sides by the integrating factor

vi. Use initial conditions to find particular solutions

3. Laplace Transform