Odinary Differential Equations (ODE)

In this post I want present some of the basic of Odinary Differential Equations, as a preparation for future post discussing solution method and numerical method for the ODE.

What is an ODE?

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives. The term “ordinary” is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.

So to show a few examples:

All of those are ODEs because only one independent variable is involved. Notice especially the third equation, even it includes high order derivatives and nonlinear functions, it’s STILL an ODE.

Order of ODE

The order of an ODE is defined as the highest order of derivatives involved. For example:

is a 3rd-order ODE.

Linear or nonlinear?

According to Wikipedia, linear differential equations are differential equations having solutions which can be added together in particular linear combinations to form further solutions. They can be ordinary (ODEs) or partial (PDEs).

In practice, this means for a n-order ODE:

The function F should be a linear combination of y, y’,y’‘… and contains no nonlinear parts like $y^2$ or $cos(y)$.

Let’s confirm our definition of Linear ODE.

1. Linear

If $y_1(x)$ and $y_2(x)$ are two solutions of this equation, then their linear combination $(ay_1(x)+by_2(x))$ will result in:

Since $ay_1'(x)=x^2y_1$ and $ay_2'(x)=x^2y_2$, so the equation stills holds, which confirmed that it’s a linear ODE.

1. Nonlinear

This time the left hand side will read: $(ay_1+by_2)^2(ay_1'+by_2')$

Which is already, obviously not equal to the original one, thus it’s a nonlinear ODE.

And that’s all for this time!

Written on June 1, 2016