## Differential equations

Differential equations are equations between unknown functions and their variables, and the derivatives of the unknown functions.

## Ordinary differential equations (ODE)

Ordinary differential equations are equations between unknown functions of one variable and the derivatives of the unknown functions.

## Partial differential equations

A partial differential equations is when the unknown function of more than one variable depends.

## Differential equations

Differential equations are equations which contains a function and derivatives of this function.

## Order

The order of the differential equations is the highest derivative of the function appearing in the equation. A first order differential equation thus includes the function and a maximum of the first derivative of the function.

## Notations for derivatives

$$\frac{dy}{dx}=\frac{d}{dx}y\left(x\right)=y\prime \left(x\right)=y\prime $$

## Direction field of the differential equation

Explicit ODE defined for each point of the XY plane, the gradient of the solution of the differential equation which passes through this point. Are put on a grid of the x / y plane tangents for the slope in the lattice point are obtained on the field direction. From the direction field one can estimate the function curve for different initial values of the solution. The equation for the field direction is obtained by transforming the differential equation.

$$y\prime =f\left(x,y\right)$$

## Initial values

The general solution of the differential equation is a family of functions parameterized by the constant C. The determination on a specific solution by specifying initial values. In order to specify a function value y _{ 0 } meant at x _{ 0 }. This can be determined from the general solution, the constant C.

${y}_{0}=\frac{b}{a}+C{e}^{-a{x}_{0}}$

The constant C is determined by:

$C=\left({y}_{0}-\frac{b}{a}\right){e}^{a{x}_{0}}$