7.1 Separable Differential Equations

One technique to solve differential equations is the method of separation of variables. Let’s look at an example:

What is the general solution to \(\displaystyle \frac{dy}{dx} = yx^{2}\)? To solve this expression we collect the variables involving \(x\) and one side of the equation, and the variables involving \(y\) on the other:

\[\begin{equation} \frac{1}{y} dy = x^{2} dx. \end{equation}\]

Now the next step is to determine the antiderivative of both sides of expression:

\[\begin{equation} \begin{split} \int \frac{1}{y} dy = \ln(y) + C. \\ \int x^{2} dx = \frac{1}{3} x^{3} + C. \end{split} \end{equation}\]

Finally since both sides are equal we can solve for the dependent variable \(y\). One thing to note: usually for antiderivatives we always include a \(+C\). For solving differential equations it is okay just to keep only one \(+C\), which usually is best on the side of the independent variable:

\[\begin{equation} \ln(y) =\frac{1}{3} x^{3} + C \rightarrow e^{\ln(y)} = e^{\frac{1}{3} x^{3} + C} \rightarrow y = Ce^{\frac{1}{3} x^{3}}. \end{equation}\]

We are in business! So here is a general technique approach to solving a differential equation via separation of variables:

1. Separate the variables on one side of the equation.
2. Integrate both sides individually.
3. Solve for the dependent variable.

If we solve this equation using separation of variables we havUsing your work above as a guide, solve this differential equation to determine a solution \(y(x)\).