wherein p0, p1 ... are power series in x, y, should satisfy the equation, it is necessary, as we find by equating like terms, that
p1 = δ0p0, p2 = δ0p1 + δ1p0, &c.
and in generalProof of the existence of integrals.
ps+1 = δ0ps + s1δ1ps-1 + s2δ2ps-2 +... + δsp0,
where
sr = (s!)/(r!) (s − r)!
Now compare with the given equation another equation
A(xyt)dF/dx + B(xyt)dF/dy = dF/dt,
wherein each coefficient in the expansion of either A or B is real and positive, and not less than the absolute value of the corresponding coefficient in the expansion of a or b. In the second equation let us substitute a series