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6.18 First-order linear equations and integrating factors

A first-order differential equation is linear if it can be written in the form:

\frac{dy}{dx}+p(x)y=q(x)

for some functions $p(x)$ , $q(x)$ . The word ‘linear’ refers to the powers of $y$ and $\frac{dy}{dx}$ which appear, not the terms in $x$ .

Integrating factor For an equation in the above form, the integrating factor is

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

we now have an integral equation:

For example, for the equation $\frac{dy}{dx}+2xy=x$ , we have $p(x)={2x}$ and therefore the integrating factor is $I(x)={e^{\int 2x\,dx}=e^{x^{2}}.}$