> [!NOTE] Definition (Separable Differential Equation) > A separable differential equation is a [[Scalar Ordinary Differential Equation|scalar ordinary differential equation]] of the form $\frac{d}{dt} x(t) = f(x(t))g(t)$with $t\in(\alpha,\beta)\subset \mathbb{R},$ $f:\mathbb{R}\to \mathbb{R}$ and $g:(\alpha,\beta)\to \mathbb{R}.$ # Properties By [[Implicit Solution to Separable Differential Equation]], $\int \frac{1}{f(\tilde{x})} \, d\tilde{x} = \int g(\tilde{t}) \, d\tilde{t}.$ **IVP**: Suppose we impose an [[Initial Value Problem|initial condition]] $x(t_{0})=x_{0}.$ By [[Constant Solution to Initial Value Problem For Separable Equation]], if $f(x_{0})={0},$ then $x(t)=x_{0}$ for all $t\in (\alpha,\beta)$ is a solution to the initial value problem. By [[Implicit Solution to Initial Value Problem for Separable Equation]], $\int_{x_{0}}^{x(t)} \frac{1}{f(\tilde{x})} \, d\tilde{x} = \int_{t_{0}}^{t} g(\tilde{t}) \, d\tilde{t},$ for all $t\in(\alpha,\beta).$