We turn now to another important method for solving first-order differential equations that rests upon the idea of “integrating both sides.” In Section 1.4 we saw that the first step in solving a separable differential equation is to multiply and/or divide both sides of the equation by whatever is required in order to separate the variables. For instance, to solve the equation
we divide both sides by y (and, so to speak, multiply by the differential dx) to get
Integrating both sides then gives the general solution .
There is another way to approach the differential equation in (1), however, which—while leading to the same general solution—opens the door not only to the solution ...