The Integrating Factor Method

The integrating factor method solves a first-order linear differential equation — any equation you can write as y′ + P(x)·y = Q(x). The trick is to multiply the whole equation by a carefully chosen function so the left side collapses into a single derivative you can integrate directly.

1. Standard form: rearrange the equation into y′ + P(x)·y = Q(x).
2. Integrating factor: compute μ = e^∫P(x)dx.
3. Multiply through by μ. The left side becomes exactly (μ·y)′.
4. Integrate both sides and solve for y, remembering the constant of integration.

Multiplying by μ = e^∫P(x)dx is designed so the product rule runs in reverse: μ·y′ + μ·P(x)·y is precisely the derivative of μ·y. That single derivative is what makes the equation integrable in one step. This is a different tactic from spotting an exact differential equation, though the two topics often appear together.

Not every first-order equation is linear. A Bernoulli equation can be substituted into linear form and then solved with an integrating factor, and a slope field lets you picture the family of solutions without solving at all.

Related lessons

• Differential Equations