Separable Differential Equations (AA HL)
A first order differential equation is separable when the \(x\) and \(y\) terms can be split apart, one side made entirely of \(x\)'s and the other entirely of \(y\)'s. Once separated, each side is integrated on its own. This page covers the method, worked examples, and the mistakes that lose the most marks. It's part of the broader Differential Equations topic.
35 questions on this sub-topic.
Separating the variables
Covered under IB syllabus reference AHL5.18. There's no formula-booklet entry for this method - it's a rearranging-then-integrating skill, so the working itself carries the marks.
General separable form
\(\dfrac{dy}{dx} = f(x)g(y)\)
Recognise this shape whenever the right-hand side factorises into a piece depending only on \(x\) and a piece depending only on \(y\).
Solving it
\(\displaystyle\int \dfrac{1}{g(y)}\,dy = \int f(x)\,dx\)
Divide both sides by \(g(y)\) first, then integrate independently - add a single constant of integration once, on either side.
Need the full syllabus wording, Euler's method, and the integrating factor too? See Differential Equations.
Worked examples
Find the general solution of \(\dfrac{dy}{dx} = \dfrac{x}{y}.\)
Worked solution
\(y\,dy = x\,dx.\) M1 A1
\(\tfrac{y^2}{2} = \tfrac{x^2}{2} + c.\) M1 A1
\(y^2 - x^2 = C.\) A1
Solve \(\dfrac{dy}{dx} = xy\) given that \(y = 2\) when \(x = 0\).
Worked solution
\(\dfrac{1}{y}\,dy = x\,dx.\) M1 A1
\(\ln|y| = \tfrac{x^2}{2} + c.\) M1 A1
\(y = A e^{x^2/2}\); \(y(0)=2 \Rightarrow A = 2.\) M1 So \(y = 2e^{x^2/2}.\) A1
On the GDC: once you have \(y = 2e^{x^2/2}\), plug in a value of \(x\) directly on the home screen to sanity-check your algebra. See the parent topic's GDC guidance for calculator-specific steps.
For the equation \(\dfrac{dy}{dx} = xy,\) separate the variables (write it with all \(y\) terms on one side and all \(x\) terms on the other).
Worked solution
Dividing by \(y\): M1
\(\dfrac{1}{y}\,dy\) A1
\(= x\,dx.\) A1
Common mistakes
- Trying to separate an equation that isn't separable. \(\dfrac{dy}{dx} + 3y = x\) can't be split into a pure-\(x\) side and a pure-\(y\) side - that shape needs the integrating factor instead, not this method.
- Forgetting the constant of integration, or adding one on each side. Only one arbitrary constant is needed once both sides have been integrated - writing \(+c_1\) on the left and \(+c_2\) on the right is unnecessary and can lead to confusion later.
- Losing the modulus sign when integrating \(\dfrac{1}{y}\). \(\displaystyle\int \dfrac{1}{y}\,dy = \ln|y| + c\), not \(\ln y\) - dropping the absolute value can cause errors when exponentiating back to find \(y\).
Ready to practise properly?
38 separable-equation questions, marked instantly like the real exam.
Quick answers
What makes a differential equation separable?
A first order equation is separable when it can be written as \(\dfrac{dy}{dx} = f(x)g(y)\), so all the \(y\) terms can be moved to one side and all the \(x\) terms to the other before integrating.
How do you solve a separable differential equation?
Rearrange into \(\dfrac{1}{g(y)}\,dy = f(x)\,dx\), integrate both sides separately, then add a single constant of integration and rearrange for \(y\) if required.