Differential Equations · step 2 of 14

Separable equations

An equation is separable when it can be written dy/dx = f(x)g(y). Then you divide by g(y), multiply by dx, and integrate both sides separately — a single arbitrary constant suffices. The step to watch is the division: any root of g(y) = 0 gives a constant solution that separation silently discards.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Separate the variablesGet all y terms with dy and all x terms with dx: dy/g(y) = f(x)dx.
Integrate both sidesAdd just one constant C, conventionally on the x side.
Solve for y if possibleExponentiate or rearrange. An implicit solution is acceptable when y cannot be isolated.
Recover the lost solutionsCheck each root of g(y) = 0 separately — these constant functions are genuine solutions.

Worked example

Solve dy/dx = xy with y(0) = 3.

Separate: dy/y = x dx (noting y = 0 is also a solution, excluded by the initial condition).
Integrate: ln|y| = x²/2 + C.
Exponentiate: y = Ae^(x²/2) where A = ±e^C.
Apply y(0) = 3: 3 = Ae⁰ = A.

Answer. y = 3e^(x²/2). Check: y′ = 3x e^(x²/2) = xy ✓.

Where marks get dropped

These are the specific errors that cost credit on separable equations questions — QED's rubric penalises each of them separately.

Losing the constant solution y = 0. Dividing by y assumes y ≠ 0, and the equilibrium must be checked separately.
Adding a constant to both sides. One constant is enough; two just merge.
Forgetting the modulus in ln|y|, then mishandling negative solutions when exponentiating.

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Separable equations — frequently asked questions

Is treating dy/dx as a fraction legitimate?

It is shorthand for the chain rule and can be fully justified. Writing ∫dy/g(y) = ∫f(x)dx is the rigorous version of the same manipulation.

What if I cannot solve for y?

Leave the answer implicit. An equation relating x and y that satisfies the ODE is a complete solution.

How do I spot separability?

Try to factor the right side into a function of x times a function of y. dy/dx = x + y is NOT separable; dy/dx = xy is.

The rest of Differential Equations

Solving and modelling with ODEs, from separable to systems. Each subtopic below has its own method, worked example and mark-losing traps.

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