## Section8.3Separable ODEs

Separable ODEs are first order ODEs with the special form:

\begin{equation} \frac{dy}{dx}=h(x)g(y).\tag{8.3.1} \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 (8.3.1) 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{8.3.2}\\ \frac{1}{g(y)}\, dy\amp =h(x)\, dx,\tag{8.3.3}\\ \int\frac{1}{g(y)}\, dy\amp =\int h(x)\, dx.\tag{8.3.4} \end{align}

### Activity8.3.1.

Solve the differential equation

\begin{equation} \frac{dy}{dx}=y\left(3x-2\right).\tag{8.3.5} \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