Key takeaway: Separation of variables solves a differential equation by moving all the y-terms with dy to one side and all the x-terms with dx to the other, then integrating both sides.
Grade 12 – AP Calculus ABAP Calculus AB Unit 7Calculus
Definition of Separation of Variables
Separation of variables works on equations where the derivative factors into a part that depends only on \(x\) and a part that depends only on \(y\). Multiplying and dividing sorts the pieces so that each variable has its own side.
Then each side is an ordinary integral. One arbitrary constant is enough, usually placed on the \(x\) side. The initial condition fixes it, and when possible the final answer is solved explicitly for \(y\).
Formula and notation
If \(\frac{dy}{dx}=p(x)\,q(y)\), then
\[ \frac{1}{q(y)}\,dy=p(x)\,dx\quad\Longrightarrow\quad \int\frac{1}{q(y)}\,dy=\int p(x)\,dx+C. \]
Any \(y\) value that makes \(q(y)=0\) gives a constant solution that division may hide.
Key facts
- Treat dy and dx as separate quantities only for the purpose of rearranging.
- Put the constant of integration in once, after integrating both sides.
- Use the initial condition right after integrating to find the constant.
- Separation of variables fails on equations like dy/dx = x + y, which cannot be factored.
Separation of Variables example
Solve \(\frac{dy}{dx}=\frac{x}{y^2}\) with \(y(0)=2\).
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Separate: \(y^2\,dy=x\,dx\).
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Integrate both sides: \(\frac{y^3}{3}=\frac{x^2}{2}+C\).
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Apply \(y(0)=2\): \(\frac{8}{3}=0+C\), so \(C=\frac{8}{3}\).
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Multiply by 3: \(y^3=\frac{3}{2}x^2+8\).
The particular solution is \(y=\sqrt[3]{\tfrac{3}{2}x^2+8}\).
Common mistake
Watch out
A student integrates \(y^2\,dy=x\,dx\) to get \(\frac{y^3}{3}=\frac{x^2}{2}\), omits \(C\), and then finds that \(y(0)=2\) leads to \(\frac{8}{3}=0\).
Why it is wrong
The contradiction signals the missing constant. Every indefinite integral brings a \(+C\), and it is the constant that lets the curve pass through the given point. Add \(C\) before using the starting value, then solve for it. Separation of variables always needs that step.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus