The integrating factor method solves a first-order linear differential equation of the form y' + P(x)y = Q(x). Multiply the equation by the integrating factor e raised to the integral of P dx, and the left side becomes a single derivative you can integrate. Learn the four steps and why the factor works.
What the integrating factor method solves
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.
The integrating factor method in four steps, from standard form to the solution.
The four steps
Standard form: rearrange the equation into y′ + P(x)·y = Q(x).
Integrating factor: compute μ = e∫P(x)dx.
Multiply through by μ. The left side becomes exactly (μ·y)′.
Integrate both sides and solve for y, remembering the constant of integration.
Why the factor works
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.
Related first-order methods
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.