Variables Separable Method

Separation of variables — the oldest and most direct method for solving a first-order equation — was introduced by Leibniz and later formalised by John Bernoulli in 1694.

Definition. A first-order differential equation is separable if it can be written as $h(y)y'=g(x)$, where the left side is the product of $y'$ and a function of $y$ alone, and the right side is a function of $x$ alone. Equivalently, an equation of the form

is called an equation with variable separable, or simply a separable equation.

Method. Rewrite the equation as

then integrate both sides independently:

where $C$ is an arbitrary constant.

Remarks.

1. There is no need to add arbitrary constants to both sides — the two constants that would arise from integrating each side separately are absorbed into the single overall constant $C$.
2. The resulting solution, carrying that one arbitrary constant, is the general solution of the differential equation.

Because the whole solving process reduces to integration, "solving a differential equation" is also called "integrating a differential equation."

Worked pattern (Example 10.11). $\left(1+x^2\right)\dfrac{dy}{dx}=1+y^2$ separates directly to $\dfrac{dy}{1+y^2}=\dfrac{dx}{1+x^2}$, integrating to $\tan^{-1}y=\tan^{-1}x+C$. Using the identity $\tan^{-1}y-\tan^{-1}x=\tan^{-1}\!\left(\dfrac{y-x}{1+xy}\right)$, this can be rewritten as $y-x=a(1+xy)$ for a renamed constant $a=\tan C$ — showing that a separated inverse-trig answer can often be tidied into an algebraic one using an inverse-trig addition/subtraction identity. …