| d | (y1′y2 − y1y2′) = P(y1′y2 − y1y2′), |
| dx |
so that we have
y1′y2 − y1y2′ = A exp. (∫ Pdx),
where A is a suitably chosen constant, and exp. z denotes ez. In terms of the two solutions y1, y2 of the differential equation having zero on the right side, the general solution of the equation with R = φ(x) on the right side can at once be verified to be Ay1 + By2 + y1u − y2v, where u, v respectively denote the integrals
u = ∫ y2φ(x) (y1′y2 − y2′y1)−1dx, v = ∫ y1φ(x) (y1′y2 − y2′y1)−1dx.
The equation
| d²y | + P | dy | + Qy = 0, |
| dx² | dx |
by writing y = v exp. (-½ ∫ Pdx), is at once seen to be reduced to d²v/dx² + Iv = 0, where I = Q − ½dP/dx − ¼P². If η = − 1/v dv/dx, the equation d²v/dx² + Iv = 0 becomes dη/dx = I + η², a non-linear equation of the first order.
More generally the equation
| dη | = A + Bη + Cη², |
| dx |