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

dydx=2xy(y>0), (1)

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

dyy=2x dx.

Integrating both sides then gives the general solution ln y=x2+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 ...

Get Differential Equations and Linear Algebra, 4th Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.