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

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Linear, Inhomogeneous Type[edit]

General Form[edit]

Solution[edit]

, where

  • is a constant and

Separable[edit]

General Form[edit]

Solution[edit]

Rearrange to get , and integrate

Bernoulli's[edit]

General Form[edit]

Solution[edit]

Substitute

Exact Equations[edit]

General Form[edit]

, with

Solution[edit]

Solution is of the form , a constant, where and

Approximation Methods[edit]

Let

Euler's Method[edit]

Euler's method with step size is given by:

.

Improved Euler's Method[edit]

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

.

Runge-Kutta Method of Fourth Order[edit]

For step size ,

, where

Few Useful Definitions · Second Order Homogeneous Ordinary Differential Equations