1. Numerical Solution of Ordinary Differential Equation

2. Ordinary Differential Equations
Equations which are composed of an unknown function and its derivatives are called differential equations.
Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
v- dependent variable
t- independent variable

3. Differential equations are also classified as to their order.
When a function involves one dependent variable, the equation is called an ordinary differential equation (or ODE). A partial differential equation (or PDE) involves two or more independent variables.
Differential equations are also classified as to their order.
A first order equation includes a first derivative as its highest derivative.
A second order equation includes a second derivative.
Higher order equations can be reduced to a system of first order equations, by redefining a variable.

4. Differential Equations
A differential equation is a relation between the independent variable (x), the dependent variable (y) and its derivatives (y’,y’’,y’’’,y(4),…). Some of these variables might be missing from the equation. Many situations in not only mathematics, but physics, engineering, biology, chemistry, economics as well as many other disciplines can be described using differential equations. Here are some examples:
Free Falling Body
Population Growth
Population Growth (limited resources)
Harmonic Oscillator
Newton’s Law of Cooling
Spread of Disease
Shape of a hanging string
Predator-Prey
Given equations like these we would like to “solve” them.

6. First Order Taylor Series Method (Euler Method)

7. Interpretation of Euler Methody2 y1 y0
x x x x

8. Interpretation of Euler Method
Slope=f(x0,y0) y1
y1=y0+hf(x0,y0)
hf(x0,y0) y0
x x x x h

9. Interpretation of Euler Methody2
y2=y1+hf(x1,y1)
Slope=f(x1,y1)
hf(x1,y1)
Slope=f(x0,y0) y1
y1=y0+hf(x0,y0)
hf(x0,y0) y0
x x x x h h

12. Runge-Kutta 4th Order Method
For
Runge Kutta 4th order method is given by
where

13. How to write Ordinary Differential Equation
How does one write a first order differential equation in the form of
Example
is rewritten as
In this case