All subjects · Mathematics · WMA14 (P4)

Separable differential equations

A differential equation links a function with its derivative. If you can separate the variables, you can integrate both sides to solve it.

Separating

If dy/dx = f(x)g(y), rearrange to get all the y terms with dy and all the x terms with dx:

∫ 1/g(y) dy = ∫ f(x) dx.

Integrate both sides and add one constant c.

General and particular solutions

The general solution contains c and describes a family of curves. A particular solution uses a given condition (such as y = 3 when x = 0) to find c.

Exponential answers

ln|y| = x2 + c gives |y| = ex2 + c = ec ex2, so y = Aex2 where A is a constant. Writing c as ln A is a neat alternative: ln y = x2 + ln A.

Worked example

Solve dy/dx = 2xy, given y = 3 when x = 0.

  1. ∫ (1/y) dy = ∫ 2x dx
  2. ln|y| = x2 + c
  3. y = Aex2; x = 0, y = 3 gives A = 3

Answer: y = 3ex2

Key idea: Separate the variables, integrate both sides with one constant, then use the condition for a particular solution.

Check yourself

  1. Find the general solution of dy/dx = 3y.
    • y = 3x + c
    • y = Ae3x
    • y = e3x + c
    • y = (3/2)y2 + c
    Show answer

    y = Ae3x. ∫ (1/y) dy = ∫ 3 dx gives ln|y| = 3x + c, so y = Ae3x.

  2. Solve dy/dx = 2xy, given y = 2 when x = 0. Find y when x = 1, to 3 significant figures.
    Show answer

    5.44. y = 2ex2. At x = 1: 2e = 5.44.

  3. Find the general solution of dy/dx = y/x, for x > 0 and y > 0.
    • y = x + c
    • y = Ax
    • y = ln x + c
    • y = Aex
    Show answer

    y = Ax. ∫ (1/y) dy = ∫ (1/x) dx gives ln y = ln x + ln A = ln(Ax).

  4. Solve dy/dx = x/y, given y = 2 when x = 1. Find y when x = 3 (with y > 0), to 3 decimal places.
    Show answer

    3.464. ∫ y dy = ∫ x dx: y2/2 = x2/2 + c. Using (1, 2): c = 3/2, so y2 = x2 + 3. At x = 3: y = sqrt(12) = 3.464.

  5. Solve dy/dx = ex - y, given y = ln 2 when x = 0. Find y when x = 1, to 3 decimal places.
    Show answer

    1.313. ex - y = ex e-y, so ∫ ey dy = ∫ ex dx: ey = ex + c. At x = 0: 2 = 1 + c, so c = 1. At x = 1: y = ln(e + 1) = 1.313.

  6. When solving a separable differential equation, how many constants of integration do you need?
    • Two, one for each side
    • None
    • One, added after integrating each side
    • One for every term
    Show answer

    One, added after integrating each side. Constants on both sides combine into one.

  7. dy/dt = -0.2y and y = 50 when t = 0. Find t when y = 10, to 2 decimal places.
    Show answer

    8.05. y = 50e-0.2t. 10 = 50e-0.2t gives t = ln 5/0.2 = 5ln 5 = 8.05.

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Last updated 7 November 2026. Spotted a mistake? Tell us.