Classifying Ordinary Differential Equations

Ordinary differential equations (ODEs) have several basic properties that you must be aware of.

One of the first important ones is its order. The order of an ODE is simply equal to the highest order derivative found in the equation. So, for example, the equation

y''' + 2y'' - 6xy' + x²y + y⁴ = sin(x)

is a third order equation, because y''' is a third order derivative, and is the highest order derivative found in the equation. Please note the important difference between terms such as y''' and y⁴; the latter is not a derivative and so is not considered when determining order of an ODE.

A second very important characteristic of an ODE is linearity. An ODE is linear if none of the y terms have exponents or involve any other "fancy" functions (like cos(y), or e^y). Squared x's are okay, but squared y's, if present, mean the equation is nonlinear. So, the equation

y''' + 2y'' - 6xy' + x²y + y⁴ = sin(x)

is nonlinear, because of the y⁴. But the equation

y'' -  x²y' + e^xy = sin(x)

is linear because the only y terms are y derivatives and y itself.

A linear equation is always capable of being arranged into the following form:

a₁(x)y + a₂(x)y' + a₃(x)y'' + ... = f(x)

That's kind of icky to look at, so let me break it down. Each term on the left contains y or a y derivative. a_n(x) are just the coefficients of each term. Note that these coefficients can be functions of x, but they can also just be constants, or even 0. There can be any number of terms, with increasing orders of derivatives, and all these terms are summed. Any terms that do not contain a y or a y derivative are written on the righthand side, which here is represented as f(x). f(x) is called the forcing function, and it can be a function of x, a constant, or 0.

This leads me to the last two properties. These last properties ONLY apply to linear ODEs. If your ODE is nonlinear, it cannot be classified any more specifically. If it is linear, further analysis must be done.

I said above that the coefficients a_n(x) can be functions of x or constants. This is another one of the properties. If ALL coefficients are constant, a linear ODE can be further classified as a constant coefficient ODE. If even one coefficient contains an x, the ODE is said to have variable coefficients. So the equation

y'' + 6y' - 11y = sin(x)

has constant coefficients, and the equation

y'' + 3xy' - 7y = 0

has variable coefficients.

Finally, like I mentioned before, the function f(x) on the righthand side of the generic linear equation listed above can be a function of x, a constant, or 0. If it is zero, the equation is said to be homogeneous. If it's not zero, the equation is nonhomogeneous. Another way to think about this is, if there is a term anywhere in the equation that does not contain y or a derivative of y, the equation is nonhomogeneous. So the equation

y'' + 5xy' + y = 7

is nonhomogeneous, while

y'' + sin(x)y = 0

is homogeneous.

So, quiz yourself: is the following equation homogeneous?

y'' + 7x²y' + y² = 0

Trick question! That equation isn't linear, so it cannot even be classified as homogeneous or nonhomogeneous. Those categories only apply to linear equations.

Okay, here's a real question. Is the following equation homogeneous?

y'' + (8/x)y' + √(x)y + 7 = 0

The answer is... no! Did I fool you? Don't get lazy and just look for a zero on the right side of the equation. There is a term (7) that does not contain y or a derivative of y, so the equation is inhomogeneous. This equation isn't shown in the standard form of a linear equation. If it were, that 7 would have been subtracted over to the righthand side, and this equation wouldn't be so deceptive.

One final question, to provide an overview of everything in this post. No tricks in this one, I promise. Tell the order of the following equation, whether it is linear or nonlinear, and, if applicable, whether it has constant/variable coefficients and is homogeneous/nonhomogeneous.

y''' + 2xy'' sin(x)y = 1/x

The answers are (highlight to read): third order, linear, variable coefficients, and nonhomogeneous.

As always, come to MASH if you want more help!

Happy calculating,
Rachel