Ordinary Differential Equations:Cheat Sheet/First Order Ordinary Differential Equations

Linear, Inhomogeneous Type

General Form

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

Solution

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

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

Separable

General Form

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

Solution

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

Bernoulli's

General Form

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

Solution

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

Exact Equations

General Form

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

Solution

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

Approximation Methods

Let $y^{\prime }=f(x,y),y(0)=y_0$

Euler's Method

Euler's method with step size $h$ is given by:

$y_{n+1}=y_n+hf(x_n,y_n)$.

Improved Euler's Method

Improved Euler's method with step size $h$ is given by:

$y_{n+1}=y_n+\frac{h}{2}[f(x_n,y_n)+f(x_{n+1},{\overline{y}}_{n+1})],{\overline{y}}_{n+1}=y_n+hf(x_n,y_n)$.

Runge-Kutta Method of Fourth Order

For step size $h$,

$y_{n+1}=y_n+\frac{h}{6}[k_1+2k_2+2k_3+k_4]$, where

• $k_1=f(x_n,y_n)$
• $k_2=f(x_n+\frac{h}{2},y_n+\frac{h}{2}{k}_1)$
• $k_3=f(x_n+\frac{h}{2},y_n+\frac{h}{2}{k}_2)$
• $k_4=f(x_n+h,y_n+h{k}_3)$