Numerical Solution of Ordinary Differential Equation http://numericalmethods.eng.usf.edu

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

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.

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.

First Order Taylor Series Method (Euler Method)

Interpretation of Euler Method y2 y1 y0 x0 x1 x2 x

Interpretation of Euler Method Slope=f(x0,y0) y1 y1=y0+hf(x0,y0) hf(x0,y0) y0 x0 x1 x2 x h

Interpretation of Euler Method y2 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 x0 x1 x2 x h h

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Runge-Kutta 4th Order Method For Runge Kutta 4th order method is given by where http://numericalmethods.eng.usf.edu

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 http://numericalmethods.eng.usf.edu

http://numericalmethods.eng.usf.edu