Separation of Variables
In short: Separation of variables cracks certain rate equations by herding all the y-stuff beside dy and all the x-stuff beside dx, so that each side can be integrated on its own.
AP Calculus ABCalculusAP Calculus AB Unit 7
What is Separation of Variables?
Separation of variables works on equations of the form \(\frac{dy}{dx} = g(x)\,h(y)\), where the right side factors into a piece with only \(x\) and a piece with only \(y\). Multiply and divide to collect all the \(y\)’s with \(dy\) on the left and all the \(x\)’s with \(dx\) on the right.
Then integrate each side separately: \(\int \frac{1}{h(y)}\,dy = \int g(x)\,dx\). Only one constant of integration is needed, and an initial condition fixes it. Finally, solve for \(y\) if possible.
How to spot it
- A differential equation is a product or quotient where x-parts and y-parts can be pulled apart, like dy/dx = x/y^2.
- A rate is described as proportional to the amount present, which gives dy/dt = ky.
- The problem asks for a particular solution that passes through a given initial point.
- A slope-field question asks for the formula behind the curve, and separation of variables can supply it.
The method
- Start separation of variables by giving each letter its own differential: \(\frac{dy}{h(y)} = g(x)\,dx\).
- Integrate both sides and add a single constant \(C\) on the right.
- Substitute the initial condition to find \(C\).
- Solve for \(y\) when you can, and state the final formula.
Separation of Variables: solved example
Problem
Solve \(\frac{dy}{dx} = \frac{x}{y^2}\) with \(y(0) = 2\).
Solution
Separate: \(y^2\,dy = x\,dx\). Integrate: \(\frac{y^3}{3} = \frac{x^2}{2} + C\).
Use \(y(0) = 2\): \(\frac{8}{3} = 0 + C\), so \(C = \frac{8}{3}\).
Multiply by 3: \(y^3 = \frac{3x^2}{2} + 8\). Hence \(y = \sqrt[3]{\frac{3}{2}x^2 + 8}\).
Try it
Question 1
A culture of yeast satisfies \(\frac{dy}{dt} = -0.5y\) with \(y(0) = 40\). Find \(y(t)\).
Show the answer
\(\frac{dy}{y} = -0.5\,dt\) gives \(\ln|y| = -0.5t + C\), so \(y = Ae^{-0.5t}\). With \(y(0) = 40\), \(y = 40e^{-0.5t}\).
Question 2
Can \(\frac{dy}{dx} = x + y\) be solved by this method?
Show the answer
No. The right side is a sum, not a product of an x-piece and a y-piece, so the variables cannot be moved to opposite sides.
Separation of Variables vs. Differential Equation
A differential equation is the problem; separation of variables is one technique for solving certain kinds of them. It only works when the right side factors into a function of \(x\) times a function of \(y\). Other equations need different tools or numerical estimates.
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