Suppose that the only functions we have come across are quotients of polynomials and that we are asked to solve the differential equation
for all x > 0, subject to the condition f(1) = 0.
If we try f(x) = P(x)/Q(x), where P and Q are polynomials with no common factors, then we find that
By comparing coefficients we can show that Q(0) = P(0) = 0, which shows that, contrary to our assumptions, both P(x) and Q(x) are divisible by x. Thus, we cannot solve equation (1) in terms of known functions. However, THE ...