Differential Equations and Linear Algebra, 4th Edition

1.4 Separable Equations and Applications

In the preceding sections we saw that if the function f(x, y) does not involve the variable y, then solving the first-order differential equation

$${\displaystyle \frac{dy}{dx}}=f(x,y)$$ (1)

is a matter of simply finding an antiderivative. For example, the general solution of

$${\displaystyle \frac{dy}{dx}}=-6x$$ (2)

is given by

$$y(x)=\int -6xdx=-3x^2+C.$$

If instead f(x, y) does involve the dependent variable y, then we can no longer solve the equation merely by integrating both sides: The differential equation

$${\displaystyle \frac{dy}{dx}}=-6xy$$ (3a)

differs from Eq. (2) only in the factor y appearing on the right-hand side, but this is enough to prevent us from using the same approach to solve Eq. (3a) that was successful with Eq. (2).

And yet, as we will see throughout the remainder of this ...