Differential Equations and Linear Algebra, 4th Edition

1.5 Linear First-Order Equations

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

$${\displaystyle \frac{dy}{dx}}=2xy(y>0),$$ (1)

we divide both sides by y (and, so to speak, multiply by the differential dx) to get

$${\displaystyle \frac{dy}{y}}=2xdx.$$

Integrating both sides then gives the general solution $\ln y=x^2+C$.

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 ...