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?
- [latex]{y}^{\prime }-4y={x}^{2}-3x+4[/latex]
- [latex]{x}^{2}y\text{'''}-3xy\text{''}+x{y}^{\prime }-3y=\sin{x}[/latex]
- [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]
You can view the transcript for “4.1.1” here (opens in new window).