AP Calculus AB / Unit 7: Differential Equations
7.6 Finding General Solutions Using Separation of Variables
Notes
Move all $y$ terms with $dy$ and all $x$ terms with $dx$, then integrate both sides.
- $\displaystyle \frac{dy}{dx}=g(x)h(y)$ becomes $\displaystyle \int\frac{dy}{h(y)}=\int g(x)\,dx$.
- Add $+C$ right after integrating.
Practice
Separating $\displaystyle \frac{dy}{dx}=xy$ gives:
Divide by $y$, multiply by $dx$.
The general solution of $\displaystyle \frac{dy}{dx}=xy$ is:
$\displaystyle \ln|y|=\frac{x^2}{2}+C$.
For $\displaystyle \frac{dy}{dx}=\frac{x}{y}$, separating and integrating gives:
$y\,dy=x\,dx$.
Ready for more? Take the Unit 7 test.