5.2. Separable differential equations

Separable first-order ODEs

If a first-order ODE can be separated so that y and x are on each side, it can be solved by integration:

(5.10)\[\begin{align} y' &= \frac{f(x)}{g(y)} \\ \int g(y)\d{y} &= \int f(x) \d{x} + c \end{align}\]

Example: Separation of variables

Given \(y' = xy\), separate the variables and solve:

(5.11)\[\begin{align} \int \frac{\d{y}}{y} &= \int x \d{x} \\ \ln y &= \frac{x^2}{2} + c_0 \\ y &= e^{x^2/2 + c_0} \\ y &= c e^\frac{x^2}{2} \end{align}\]

Note that here, we are being careful to denote the redefinition of the integration constant (\(c = e^{c_0}\)). This detail may be glossed over at times.