2.2 | Separable Equations


If the right-hand side of the equation
$$
\frac{dy}{dx} = f(x,y)
$$
can be expressed as a function \(g(x)\) that depends only on \(x\) times a function \(p(y)\) that depends only on \(y\), then the differential equation is called separable.
Method for Solving Separable Equations
To solve the equation
$$
\frac{dy}{dx} = g(x)p(y)
$$
multiply by \(dx\) and by \(h(y):= 1/ p(y)\) to obtain
$$
h(y) \; dy = g(x) \; dx.
$$
Then integrate both sides:
$$
\begin{aligned}
\int h(y) \; dy & = \int g(x) \; dx \\
H(y) & = G(x) + C
\end{aligned}
$$
where we have merged the two constants of integration into a single symbol \(C.\) The last equation gives an implicit solution to the differential equation.

E 2.2 Exercises