Integrating factor

Linear ODE dy/dx + P(x)y = Q(x). Multiply by μ(x) = e^{∫P dx} to make LHS exact.

Linear non-separable ODEs use the integrating factor method. After multiplying by μ, the LHS becomes d/dx(μy). Integrate to find μy; divide by μ for y.

Worked examples

dy/dx + 2y = e^x: P = 2, μ = e^2x. d/dx(e^2xy) = e^3x; y = e^x/3 + Ce^−2x.

x dy/dx + y = x²: rearrange as dy/dx + y/x = x; μ = x; d/dx(xy) = x²; y = x²/3 + C/x.

Initial conditions fix C.

Frequently asked questions

How do I get the IF formula?

From requiring d/dx(μy) = μ(dy/dx + Py). This forces μ'/μ = P, giving μ = e^{∫P dx}.

Why no constant in ∫P dx?

Any constant multiplies μ and cancels. Pick the simplest IF.