Integrating Factor

Consider the first order differential equation

\( \frac{dy}{dx}+ Py = Q \)

where P and Q are functions of \(x \) only, and the coefficient of the \(\frac{dy}{dx} \) term is 1. This is called the standard form.

For a standard form differential equation, the integrating factor is

\( I = e^{\int P\, dx} \quad (assuming\ \int P \ can\ be\ found) \)

We multiply each term of the standard form differential equation by the integrating factor I to create an exact differential equation

\( I \frac{dy}{dx}+ IPy = IQ \)

Then integrating both sides, gives

\begin{align} \int \left(I\frac{dy}{dx}+IPy \right)dx & = \int IQ \ dx \\ Iy & = \int IQ\ dx \end{align}

Since I and Q are functions of \(x\) we can now integrate the RHS in the usual manner

Example

Find the general solution of the differential equation

\begin{align} x\frac{dy}{dx}+2y & = \frac{e^x}{x^3} \\ \\ \end{align}

First, divide by \(x\) to put the equation into standard form

\( \frac{dy}{dx}+\frac{2}{x}y = \frac{e^x}{x^4} \)

Then find the integrating factor

\begin{align} I & = e^{\int P} \\ & = e^{\int \frac{2}{x}} \\ & = e^{2ln(x)} \\ & = e^{ln(x^2)} \\ & = x^2 \end{align}

Now multiply the standard form equation by the integrating factor

\begin{align} x^2\frac{dy}{dx}+2xy = x^2e^x \end{align}

Then integrate both sides

\begin{align} \int \left(x^2\frac{dy}{dx}+2xy \right)dx & = \int x^2e^x \ dx \\ x^2y & = \int x^2e^x \ dx \\ x^2y & = e^x(x^2-2x+2) + c\\ \end{align}