dy/dx + yP = Q,
where P, Q are functions of x only; this is known as the linear
equation, since it contains y and dy/dx only to the first degree. If
fPdx = u, we clearly have
d /dy \
--(ye^u) =e^u ( -- + Py) = e^u Q,
dx \dx /
so that y = e^-u(fe^u Qdx + A) solves the equation, and is the only
possible solution, A being an arbitrary constant. The rule for the
solution of the linear equation is thus to multiply the equation by
e^u, where u = fPdx.