How to Classify Differential Equations

Calculus Tutorial

How to Classify Differential Equations


Differential equations can be classified into a myriad of different groups, and these can be hard to keep track of. However, identifying these qualities are critical in developing a solution. As such, it is a crucial skill to be able to identify the classifications of a differential equation. The classifications are as follows:

1. ODE vs. PDE

An ordinary differential equation is one that involves only ordinary derivatives (derivatives of a single-variable function), and a partial differential equation is one that involves partial derivatives (derivatives of multivariable functions).

2. Order

The order of a differential equation is equal to the order of the highest derivative involved in the equation. For instance, if the highest order derivative is a first derivative, then the equation is said to be first order. If the highest order derivative is the second derivative, then the equation is said to be second order. And so on.

3. Linear vs. Nonlinear

In layman’s terms, a linear differential equation is one that doesn’t have any weird stuff. Linear equations are ones where the function and its derivatives are raised to the first power only, and are not nested inside of any trigonometric functions. The function and its derivatives can have coefficients which are themselves functions of x. If y or its derivatives are raised to a power or nested inside a trig function, then the equation is said to be nonlinear.

These are the basic classifications for differential equations.

Sample Problem

Classify the following differential equation:

y” -2xy’ +2y = 0

First-Order Linear ODE

Second-Order Linear PDE

Second-Order Nonlinear ODE

First-Order Nonlinear ODE

Second-Order Linear ODE


Take the equation one step at a time. Examine each of the aspects of the equation to fully classify it.

1. ODE vs. PDE

The equation has an independent variable x and a dependent variable y. There is only one independent variable, so the derivatives are ordinary. Therefore, the equation is an ODE.

2. Order

The highest derivative in the equation is the second derivative of y (y”). Therefore, the equation is second-order.

3. Linear vs. Nonlinear

Looking at the equation, we can see that there is no “weirdness” going on. All of the y-terms and derivatives are to the first power and are not nested within any abstract functions. The y-terms and derivatives are multiplied by x-term coefficients, but this does not negate linearity. Therefore, the equation is linear.

Bringing everything together– this equation is a Second-Order Linear ODE.

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