Differential equations

A differential equation links a function with its derivatives. They model how things change: populations, cooling, springs and circuits.

Key facts

  • Separate the variables: get all y on one side, all x on the other, then integrate.
  • dy/dt = ky has solution y = Ae^(kt) (growth or decay).
  • Use the initial condition to find the constant.
  • Second order (university): ay″ + by′ + cy = 0 uses the auxiliary equation am² + bm + c = 0.

Worked examples

Example 1

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

  1. (1/y) dy = 2x dx
  2. Integrate: ln y = x² + c y = Ae^(x²)
  3. y = 3 at x = 0 gives A = 3. y = 3e^(x²)

Answer: y = 3e^(x²)

Example 2

Solve y″ − 5y′ + 6y = 0.

  1. m² − 5m + 6 = 0 (m − 2)(m − 3) = 0
  2. Two real roots, so y = Ae^(2x) + Be^(3x)

Answer: y = Ae^(2x) + Be^(3x)