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 ...

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