Definitions and equations with separable variables

What is a differential equation?

A differential equation is a relationship between an unknown function y(x) and its derivatives y'(x), y''(x), etc. The order of the equation is that of the highest-order derivative appearing in it. Solving a differential equation means finding all functions y that satisfy this relationship over a given interval.

Equations with separable variables

A first-order equation is said to have separable variables if it can be written in the form: g(y) * y' = f(x) We then separate the variables: g(y) dy = f(x) dx, and integrate each side separately.

Example: Solve y' = x * y (where y is non-zero). We write dy/y = x dx, then integrate: ln|y| = x²/2 + C Hence y = K * exp(x²/2), where K = ± exp(C), K being any real number (y = 0 is also a solution).

Common pitfall

Dividing by y assumes y is non-zero: one must always check whether y = 0 (or any root of g) is a solution ‘lost’ during separation, and then reintegrate it into the family of solutions.

Initial condition

A first-order differential equation has an infinite number of solutions, depending on a constant C. An initial condition y(x₀) = y₀ allows us to determine a unique solution from this family: this is a Cauchy problem.