gradeup
← Unit 7: Differential Equations

Separation of Variables & Exponential Growth/Decay

Separable differential equations can be solved by algebraically separating x and y terms to opposite sides, then integrating both sides. The most common application is exponential growth/decay, dy/dt = ky.

▶Khan Academy: AP Calculus ABKhan Academy — opens in a new tab

Solving by separation of variables

If dy/dx = f(x)g(y), rewrite as dy/g(y) = f(x)dx (separating all y-terms with dy on one side, all x-terms with dx on the other), then integrate both sides and add a constant of integration. Solve for y explicitly if possible, and use an initial condition to find the specific constant.

Worked Example

Solve dy/dx = ky (k constant), given y(0) = y₀.

  1. Separate: dy/y = k dx.
  2. Integrate both sides: ln|y| = kx + C.
  3. Exponentiate: y = e^(kx+C) = e^C · e^(kx) = A·e^(kx), where A=e^C is a new constant.
  4. Apply y(0)=y₀: y₀ = A·e^0 = A, so A=y₀.

Answer: y = y₀·e^(kx) — the standard exponential growth/decay model.

Key terms (1)
Separable equation
A differential equation that can be rewritten with all y-terms (and dy) on one side, all x-terms (and dx) on the other.

Practice Quiz

Question 1 of 2Score so far: 0/0

A population grows according to dP/dt = 0.05P, with P(0) = 200. What is P(t)?