7.1 Second-order linear equations

When working with differential equations, usually the goal is to find a solution. In other words, we want to find a function (or functions) that satisfies the differential equation. The technique we use to find these solutions varies, depending on the form of the differential equation with which we are working. Second-order differential equations have several important characteristics that can help us determine which solution method to use. In this section, we examine some of these characteristics and the associated terminology.

Homogeneous linear equations

Consider the second-order differential equation

Notice that y and its derivatives appear in a relatively simple form. They are multiplied by functions of x , but are not raised to any powers themselves, nor are they multiplied together. As discussed in Introduction to Differential Equations , first-order equations with similar characteristics are said to be linear. The same is true of second-order equations. Also note that all the terms in this differential equation involve either y or one of its derivatives. There are no terms involving only functions of x . Equations like this, in which every term contains y or one of its derivatives, are called homogeneous.

Not all differential equations are homogeneous. Consider the differential equation

The $x^2$ term on the right side of the equal sign does not contain y or any of its derivatives. Therefore, this differential equation is nonhomogeneous.

In linear differential equations, $y$ and its derivatives can be raised only to the first power and they may not be multiplied by one another. Terms involving $y^2$ or $\sqrt{y^{\prime }}$ make the equation nonlinear. Functions of $y$ and its derivatives, such as $\text{sin}y$ or $e^{y^{\prime }},$ are similarly prohibited in linear differential equations.

Note that equations may not always be given in standard form (the form shown in the definition). It can be helpful to rewrite them in that form to decide whether they are linear, or whether a linear equation is homogeneous.