Separable Equations

These are differential equations that, when rearranged, the derivative $\frac{dy}{dx}$ can be arranged in the form $f(x)g(y)$. I.e:
$\frac{dy}{dx}=f(x)g(y)$

How to separate variables:

Method 1:

• Solve for the derivative and you better get something like $xy$.

Method 2:

1. Check for values of $y$ that make $g(y)=0$. These correspond to constant solutions.
2. Rewrite the differential equation in the form $\frac{dy}{g(y)}=f(x)dx$.
3. Integrate both sides of the equation.
4. Solve the resulting equation for $y$, if possible!
5. If an initial condition exists, solve for the constant.