Q.If $\sin x$ is the integrating factor of the linear differential equation $\dfrac{dy}{dx}+Py=Q$, then $P$ is

A first-order differential equation is linear if it can be written as

where $P$ and $Q$ are functions of $x$ only (or constants) — crucially, there is no product of $y$ with $\dfrac{dy}{dx}$, and $y$ and its derivative occur only to the first power. (The mirror form $\dfrac{dx}{dy}+Px=Q$, with $P,Q$ functions of $y$ only, is used when the equation is more naturally linear in $x$.)

Derivation of the integrating factor. Consider first the associated homogeneous equation $\dfrac{dy}{dx}+Py=0$. Separating variables and integrating gives $y\,e^{\int P\,dx}=C$. Differentiating $y\,e^{\int P\,dx}$ using the product rule shows

whenever $y$ satisfies the original (non-homogeneous) equation — the left side collapses to an exact derivative. The quantity

is called the integrating factor.

Solution formula. Multiplying the linear equation through by the integrating factor and integrating both sides with respect to $x$ gives the closed-form general solution

For the $x$-dependent-variable mirror form, the analogous solution is $x\,e^{\int P\,dy}=\int Q\,e^{\int P\,dy}\,dy+C$.

Working steps. …