## Section9.3Separable ODEs

Separable ODEs are first order ODEs with the special form:

\begin{equation} \frac{dy}{dx}=h(x)g(y).\label{eqn-separable}\tag{9.3.1} \end{equation}

where $h(x)$ and $g(y)$ are arbitrary functions, i.e. the $x$ and $y$ dependencies and be separated into the product of two functions. A solution will exist wherever these functions are sufficiently smooth and $g(y)$ has no values of $y$ for which the function is zero.

##### Method.

First, write (9.3.1) as a differentials equation by multiplying through by $dx\text{.}$ Then divide by $h(x)$ so that all the $y$-dependence is on one side of the equation and all the $x$-dependence is on the other side of the equation. Finally, integrate both sides of the equation.

\begin{align} dy\amp =h(x)g(y)\, dx,\tag{9.3.2}\\ \frac{1}{g(y)}\, dy\amp =h(x)\, dx,\tag{9.3.3}\\ \int\frac{1}{g(y)}\, dy\amp =\int h(x)\, dx.\label{eqn-dsep}\tag{9.3.4} \end{align}
###### Activity9.3.1.

Solve the differential equation

\begin{equation} \frac{dy}{dx}=y\left(3x-2\right).\label{eqn-sepex}\tag{9.3.5} \end{equation}
Since we are doing an indefinite integral, it is important for us to remember, in the second to last step, to add the unspecified constant $C$ to one side of the equation. If we are given an initial condition such as $y(0)=7\text{,}$ then we can use this to find the value $C=\ln 7\text{.}$ The final answer is