Linear integrating factor
Turn a first-order linear equation into one product derivative.
Linear integrating factor is one of 2 first order formulas in the differential equations section of this library, and it is used at university level.
Why linear integrating factor works
Multiply the whole equation by μ and look at the left side: it becomes μy' + μpy. Since μ was built so that its own derivative equals pμ, that expression is exactly the product rule applied to μy. So the left side folds into one derivative, and integrating both sides is then a single step.
What each symbol means
$\mu$ is the integrating factor.
Linear integrating factor: when it holds
Write the equation with coefficient $1$ on $y'$ before computing $\mu$.
When it stops applying
The method needs the equation to be linear in y, so y' + y² = x is out of reach because of the squared term. It also needs p to be continuous on the interval you care about: with p = 1/x, the solution cannot be continued across x = 0.
Linear integrating factor: a worked example
After multiplication, $(\mu y)'=\mu q$, then integrate and solve for $y$.
The mistake to avoid
What people do: Students compute the integrating factor while the equation still has a coefficient in front of y'.
Why it goes wrong: The formula for μ assumes y' stands alone. Reading p off an equation like xy' + y = x² without dividing first gives the wrong exponent and the left side never folds up.
Do this instead: Divide through by the leading coefficient first. Turning xy' + y = x² into y' + y/x = x makes p = 1/x, so μ = x, and the left side becomes the derivative of xy.
Linear integrating factor: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\mu$ is the integrating factor.
- Check the conditions before substituting. Write the equation with coefficient $1$ on $y'$ before computing $\mu$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most differential equations slips.
Where this formula fits
- Subject
- Differential Equations formulas — 8 entries in this library
- Topic
- First order
- Level
- University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where linear integrating factor comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Integration Techniques — the lesson behind this formula: choose substitution, parts, or simplification from structure.
- Calculus Calculator — check your substitution and the value it produces.
- Study differential equations — the subject guide that explains the ideas these formulas compress.
- Differential Equations Practice — questions that make you retrieve the formula instead of recognising it.
- All 8 differential equations formulas — the full grouped reference, or the complete formula library.
Questions about linear integrating factor
Do I need a constant of integration inside μ?
No. An extra constant would multiply μ by a fixed number, and that factor cancels from both sides of the equation, so setting it to zero costs nothing.
How do I recognize a first-order linear equation?
Look for y and y' each appearing to the first power, with no products of the two and no y inside another function. The coefficients may be any functions of x, but y itself must appear plainly.
Can an equation be both separable and linear?
Yes, and then either method works. y' + 2y = 0 separates easily and also has μ = e^(2x), and both routes deliver y = Ce^(−2x).
What happens after I integrate both sides?
You are left with μy equal to the integral of μq plus a constant, so the last step is dividing by μ. For y' + 2y = 4 that gives y = 2 + Ce^(−2x), which you can check by substitution.