# ordinary differential equations

Please also explain when we should use ordinary derivatives and when we should use partial derivatives.

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The difference between ordinary differential equations, which we often refer to as ODEs, and partial differential equations, which we often refer to as PDEs, is that ODEs have one independent variable and PDEs have more than one. The theory of ODEs is very well worked out. PDEs are much harder to work with and a lot of research is still being done to determine when a PDE can be solved, when the solution is unique, and to find numerical (computer) methods for approximating solutions.

Partial differential equations are classified into one of three types: parabolic, elliptic, or hyperbolic. The classic example of a parabolic equation is the heat equation. The classic example of an elliptic equation is the Laplace equation. The classic example of a hyperbolic equation is the wave equation.

1 Introduction

A differential equation is one with one or more derivatives in it. We classify differential equations as ordinary or partial differential equations according to the number of independent variables. An ordinary differential equation has one independent variable. A partial differential equation as more than one independent variable, hence it contains partial derivatives. We can further classify both ordinary and partial differential equations. Both ODEs and PDEs are very important in physics, engineering, mathematical biology, and other fields.

Mathematicians study these equations to determine (a) if a solution exists, (b) if the solution is unique, (c) if the solution depends continuously on the data, and (d) to find ways to solve or approximate the equations.

2 Ordinary Partial Differential Equations

If an ODE only contains first derivatives, we call it a first order differential equation. If it contains higher order derivatives, we call it a second order, third order, etc. equation depending on the highest order derivative in the equation. First order equations are nice to work with. Sometimes we rewrite higher order ODEs as systems of first order ODEs in order to be able to solve them more easily. ODEs may be linear or nonlinear. Linear equations are not too difficult to solve. Nonlinear equations are a lot more difficult, and sometimes the solution doesn't have much meaning to us. We can use a field of math called nonlinear dynamics to graphically determine the behavior of the ODE and its solution even if we don't know the solution. This approach gives us a great deal of useful

information.

2.1 Examples

The simplest examples of ODEs have to do with periodic motion such as a simple harmonic oscillator (spring) or a pendulum. Much more complicated things can be modeled as oscillators. For example, the firing of neurons in the brain can be viewed as an oscillation. In studying Parkinson's disease, the neurons are viewed as oscillators. Since there are many neurons involved, we write large systems of ordinary differential equations to represent the behavior of neurons. Mathematicians and physicists are studying ways to represent neurons with ODEs and how the ODEs behave to figure out how much current should be applied in brain pacemakers and in what order the electrodes should be stimulated in order to treat or cure Parkinson's.

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This is a huge subject. I hope the information I am providing here is helpful to you.

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