Originally Posted by

**BobP** There are an infinite number of solutions aren't there ?

Just solve as a first order ode in $\displaystyle A(x)$.

Rewrite as

$\displaystyle A'(x) - B'(x)A(x) = B(x)B'(x) $,

multiply by the integrating factor

$\displaystyle e^{-B(x) }$

and the equation can be written as

$\displaystyle (A(x)e^{-B(x)})' = B(x)B'(x)e^{-B(x)}$

so

$\displaystyle A(x)e^{-B(x)} = \int B(x)B'(x)e^{-B(x)} dx $

and

$\displaystyle A(x) = e^{B(x)}\int B(x)B'(x)e^{-B(x)} dx $.

Now just choose something convenient for $\displaystyle B(x) $.

An easy choice would be $\displaystyle B(x) = x $ after which an integration by parts would lead to $\displaystyle A(x) = Ce^{x} - x - 1 $.