Earlier we have seen how to find roots of equation, Integration using MATLAB software.

Now moving further ahead in this post we will see how to find Ordinary Differential Equations (ODE) using MATLAB software.

In Engineering and science, the differential equations are used to explain the various phenomena and fundamental laws. Differential equations describe the changes in the state of the system.

When a differential equation contains all the derivatives with respect to single variable, then it is called as an 'Ordinary Differential Equation'.

### Engineering Applications of ODE:

The basic law of heat transfer which is known as Fourier law is given by an equation

q=(-kA)*(dT/dx)

where q is the rate of heat transfer, k is the coefficient of thermal conductivity, A is the area of cross section and (dT/Dx) is the temperature gradient.

One more application we can have is the Faraday's law of Electromagnetic Induction. According to the law, the induced voltage in a coil is given by the equation

V=N*(do/dt)

where V is the induced voltage, N is the number of turns in the coil and d0(phie)/dt ia the rate if change of flux.

The various methods to solve the problems on ODE are as follows:

- Taylor's Series Method
- Euler's Method
- Euler's Predicator-Corrector Method
- Range-Kutta 2nd order mothod
- Range-Kutta 4th order method

Euler's Method

Euler's Predictor-Corrector method

Range-Kutta 2nd order

Range-Kutta 4th order

Adam-Bashforth Formula

Milne Method

Milne's Formula

Euler's Method-another way

Range-Kutta 2nd order-another way

Range-Kutta 4th order-another way