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Integrating Factor Method

For linear equations the trick is multiplying by a factor that collapses the left side into a single perfect derivative.

Standard linear form

A first-order linear equation can be written as y prime plus p of x times y equals q of x. Linear means y and its derivative appear only to the first power, with no y squared, no y times y prime, and no sines of y. Recognizing the form matters because linear equations always yield to the same mechanical method.

y' + p(x)\,y = q(x)

The integrating factor

Multiply the whole equation by mu equal to e to the integral of p. This choice makes the left side exactly the derivative of mu times y by the product rule, since mu prime matches p times mu. The equation then reduces to the derivative of mu y equals mu q, ready for one integration.

\mu(x) = e^{\int p(x)\,dx}

Worked example

Solve y prime plus 2y equals 6. The factor mu is e to the 2x, so the derivative of e to the 2x times y equals 6 e to the 2x. Integrating gives e to the 2x times y equals 3 e to the 2x plus C, and dividing by mu yields y equals 3 plus C e to the negative 2x, which visibly settles toward the equilibrium 3.

y = 3 + Ce^{-2x}

Flashcard

y' + p(x)\,y = q(x)

Standard form of a linear first-order equation

Every linear first-order equation can be arranged this way with coefficient 1 on y prime. Getting into this form is the required first step before applying the integrating factor.

Check your understanding

What is the integrating factor for the linear equation y prime plus 3x squared times y equals x?

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