Find all functions which are continuous on the positive real axis and satisfy the equation

One such function is

We differentiate both sides of the equation the functions we are looking for must satisfy,

Since we know we can solve the original equation for ,

Substituting this into the first equation we then have

Now this is a differential equation of the form so by Theorem 8.3 we know the solutions for any given initial condition exist and are unique. Taking we have that the given function is the only one satisfying both conditions of continuity and the given equation