← Unit 7 Guide

Separation of Variables

Slope fields let you see a solution; Euler's method approximates one. This is the first tool that gets you the solution exactly — an actual formula for y. It only works for one special shape, and the whole method is really a single move followed by integrating.
If a differential equation can be rearranged so that everything with y sits with dy and everything with x sits with dx, you can integrate both sides separately and solve for y. That's the entire idea: get the variables apart, then integrate. The art is recognizing when it's possible and handling the constant of integration cleanly.

The shape that separates

Separation works only when the right-hand side factors into an x-part times a y-part. If it does, you can divide the y-part over to the dy side and you're ready to integrate. If it can't be factored that way — most often because it's a sum — you need a different method.

Can you separate it?

Spotting separability is its own skill — and a tested one. For each equation, decide whether the right side factors into (an x-part)(a y-part). Watch for sums that don't factor, and products hiding inside things you can pull apart.

Build it & solve it

The whole method is four moves: separate, integrate both sides, solve for y, then apply the initial condition. Build your own equation — pick an x-part, a y-part, and a starting value — and watch the method solve it, then thread your exact solution through its slope field. Switch to Test me to choose each step yourself.

x-part  g(x)
y-part  h(y)
initial value  y(0)
slope field your exact solution
finish all four steps to draw the solution over its field
the particular solution

The ideas everything else builds on

The separable shape

The method needs the right side to factor — an x-part times a y-part. A sum almost never qualifies.

Separate, then integrate

Divide the y-part across to the dy side, then integrate both sides. That single move is the heart of the whole technique.

Only one +C

Integrating both sides produces a constant on each — but combine them into a single +C on the x-side. Two constants is a red flag.

The condition pins it down

The general solution carries an unknown constant. The initial condition is the one point that fixes it, giving the particular solution.

Your turn

Six checks across the whole method — recognizing the shape, separating and integrating, solving for y, and pinning the constant with a condition. Try each before revealing.

Quick quiz

Six fast multiple-choice checks across the whole lesson. Pick an answer to see whether it's right and why — then work through all six.

Question 1 of 6
Score: 0 / 0

Common mistakes & exam tips

Dropping the +C

The instant you integrate, a constant appears. Leave it off and the rest of the problem is wrong — and on a free-response it's an immediate point lost. Write +C the moment you integrate.

Carrying two constants

Both sides produce a constant, but they merge into one. Put a single +C on the x-side and move on — two separate constants signals a misunderstanding.

Stopping at the general solution

If the problem gives an initial condition, you're not done at y = A·e^(…). Apply the point to solve for the constant and report the particular solution.

Mishandling e^C

When you exponentiate ln|y| = … + C, the e^C becomes a single new constant (call it A). Don't drag the C along inside the exponent.

Trying to separate a sum

dy/dx = x + y can't be separated — no division pulls the variables apart. Separation needs a product; a sum means a different method entirely.

Forgetting the absolute value

Integrating dy/y gives ln|y|, not ln y. It usually washes out into ±A when you solve for y, but the careful step is ln|y|.

On the AP exam

Separation is exactly how the standard models are derived. Next, Growth Models uses it to turn “rate proportional to amount” into exponential growth, and the logistic equation into its S-curve. Or head back to the Unit 7 Guide.