Basics of Differential Equations: Learn It 2

Order of Differential Equations

When working with differential equations, we need a way to categorize and describe them. The most fundamental characteristic is the order of the equation.

order of a differential equation

The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.

Here are some examples to illustrate:

  • [latex]y' = 2x[/latex] is first-order (highest derivative is [latex]y'[/latex])
  • [latex]y'' - 3y' + 2y = 0[/latex] is second-order (highest derivative is [latex]y''[/latex])
  • [latex]y''' + xy' = \sin(x)[/latex] is third-order (highest derivative is [latex]y'''[/latex])

Understanding the order helps us choose appropriate solution methods and tells us important information about the nature of the solutions we can expect.

What is the order of each of the following differential equations?

  1. [latex]{y}^{\prime }-4y={x}^{2}-3x+4[/latex]
  2. [latex]{x}^{2}y\text{'''}-3xy\text{''}+x{y}^{\prime }-3y=\sin{x}[/latex]
  3. [latex]\frac{4}{x}{y}^{\left(4\right)}-\frac{6}{{x}^{2}}y\text{''}+\frac{12}{{x}^{4}}y={x}^{3}-3{x}^{2}+4x - 12[/latex]

Watch the following video to see the worked solution to the example above.

You can view the transcript for “4.1.1” here (opens in new window).