Interactive differential equations

One multiplier,
two revelations.

An integrating factor is a function chosen so the equation becomes the derivative of something already known how to integrate.

Algebra and geometry agreeIn a linear ODE it creates a product derivative. In a differential form it creates a potential whose level curves are the solutions.

Linear equation

Reverse the product rule

y′ + P(x)y = Q(x), μ(x) = exp(∫P(x)dx)

Because μ′ = Pμ, multiplying gives μy′ + μPy = (μy)′. One integration then yields μy = ∫μQ dx + C.

The integration constant hidden inside ∫P only scales μ and can be discarded.

Differential form

Test exactness

For M dx + N dy = 0, exactness means Mᵧ = Nₓ. Then there is a potential F with Fₓ = M and Fᵧ = N, so solutions are F = C.

Repair

Make cross-partials agree

Seek μ so (μM)ᵧ = (μN)ₓ. If (Mᵧ − Nₓ)/N depends only on x, then μ(x) = exp(∫[(Mᵧ − Nₓ)/N] dx).

Domain

Never cross a zero silently

Multiplying by μ preserves solution paths only where μ is finite and nonzero. A factor such as μ = x requires separate domains x > 0 and x < 0.

Resonance

Watch the exceptional parameter

In y′ + py = eˣ, the formula eˣ/(p+1) fails at p = −1. The correct particular solution gains a factor x: xeˣ.

Decision guide

Classify before integrating

If the equation is y′ + Py = Q, use μ = e∫P.
If it is Mdx + Ndy = 0, test Mᵧ = Nₓ first.
If not exact, test whether a μ(x) or μ(y) criterion collapses to one variable.
After solving, differentiate the answer back into the original equation.