Solving by separation of variables

Ordinary differential equations: Separable differential equations

Solving by separation of variables

Reflecting on the previously discussed method of integrating a mathematical function, you may wonder if there are more first-order ODEs that you can rewrite to an equality of differentials. The answer to this question is affirmative.

Solving a separable differential equation The general solution of \[\frac{\dd y}{\dd t}=\frac{f(t)}{g(y)}\] satisties the equality \[G(y)=F(t)+c\] where \(F(t)\) is an antiderivative of \(f(t)\), \(G(y)\) is an antiderivative of \(g(y)\) and \(c\) is a constant.

Let \(f(t)\) and \(g(y)\) be functions with antiderivatives \(F(t)\) and \(G(y)\), respectively. The differential equation \[\frac{dy}{dt} = \frac{f(t)}{g(y)}\] can be rewritten in differential form as \[g(y)\,\dd y = f(t)\,\dd t\] and is then equivalent to \[d\bigl(G(y)\bigr) = d\bigl(F(t)\bigr)\] It follows that \[G(y)=F(t)+c\] for some constant \(c\).

Implicit and explicit solutions This theorem does in general not provide a solution of the ODE in explicit form \(y(t)=\ldots\), but only a relationship between the variables \(y\) and \(t\). Sometimes you have luck and can derive an explicit solution from the implicit solution of ODE.

Naming The first important step in solving the ODE is the rewriting of the differential equation in differential form \(g(y)\,\dd y = f(t)\,\dd t\) with separate expressions in \(y\) and \(t\). Therefore the solution strategy is known as solving by separation of variables and one speaks of a separable differential equation.

Some examples will illustrate how to solve differential equations by separation of variables.