The next simplest case is when the rate of change $\frac{dy}{dx}$ can be factored in a way that separates the dependance of $x$ and $y$. In particular,
| A differential equation is called separable if it's of the form $$\frac{dy}{dx}=f(x)g(y).$$ |
If $g\ne 0$, we can cross-multiply to get $$\frac{dy}{g(y)} = f(x)\, dx$$ and then integrate both sides. This yields $$G(y) = F(x) + C,$$ where $F$ and $G$ are anti-derivatives of $f$ and $1/g$, respectively. As always, we need to plug in the value of one point to determine $C$.
| Example: Find the
solutions of $$ \frac{dy}{dx}=xy^2+x.$$ Solution: The equation is separable, since the rate of change $\frac{dy}{dx}$ can be written as $f(x)g(y)=x(y^2+1)$. Notice that $y^2+1\ne0$. Hence, $$ \frac{dy}{y^2+1}=x\,dx $$ We integrate both sides $$ \int\frac{dy}{y^2+1}=\int x\,dx $$ to get $$ \arctan(y)=\frac{x^2}{2}+C, $$ or, $$ y=\tan\left(\frac{x^2}{2}+C\right). $$  |