Linear Differential Equations

A differential equation of the form dy/dx + Py = Q
is called a linear differential equation, where P and Q, are functions of x( but not of y) or constants.

In such case, multiply both sides by e∫Pdx
e∫Pdx[dy/dx + Py] = Qe∫Pdx

d[ye∫Pdx]/dx = Q·e∫Pdx

Integrating both sides we get
ye∫Pdx = ∫Qe∫Pdxdx + C.
e∫Pdx is called the integrating factor.

Solution is y×[I.F.] = ∫Q[I.F.]dx + c