(Moved from Differential Equations, not sure where exactly this belongs)

Suppose I have an odd, increasing function h with h(0)=0 and an unknown increasing function f(D), f(0)=0.

Let:

$\displaystyle \phi(h(f(D))*h(f(D))*h'(f(D))*f'(D)=D$

where $\displaystyle \phi$ is the standard normal pdf

From the above equation, we can see that D spans the real line and the when the RHS is > 0 so is the LHS and vice versa.

But, I can rewrite the above equation as:

$\displaystyle \frac{-\partial\phi(h(f(D))}{\partial{D}}=D$

$\displaystyle \implies 0.5*D^2=-\phi(h(f(D)) $

which does not make sense since the LHS is > 0 and the RHS is < 0