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MathidiaMathematics Academy
Differential equations
Lesson 51 of 75
Introduction

Separable equations

Objective. Solve separable initial-value problems.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A separable equation is a first-order ODE that can be written as dy/dx = f(x)g(y), so all y-terms move to one side and all x-terms to the other before integrating. Rearrange to (1/g(y)) dy = f(x) dx, integrate both sides, then use the initial condition to fix the constant. For example dy/dx = xy gives ∫(1/y) dy = ∫x dx, so ln|y| = x²/2 + C, hence y = Ae^(x²/2). Always check y = 0 is not a lost solution when dividing by g(y), and substitute the initial value last to pin down A exactly.

Worked example

Solve the separable equation dy/dx = 2xy with y(0) = 3, giving y as a function of x.

2 … 0
Illustrated card showing 2 … 0
  1. 1Separate: (1/y) dy = 2x dx
  2. 2Integrate: ln|y| = x² + C
  3. 3Apply y(0)=3: ln3 = C
  4. 4Exponentiate: y = 3e^(x²)

Answer: y = 3e^(x²)

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Common mistake. Dividing by g(y) without checking g(y) = 0 silently discards the constant solution y = 0; for dy/dx = y², y ≡ 0 solves it but vanishes once you divide by y². Always state it separately.

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