Deriving a demand function from a specific utility function

Question

Let's say I have a utility function of the form $Ax^b + Cx^d$. Now I would like to find the consumption depending on the price for one unit of good $x$. This means for any given $p$ I would maximize $Ax^b + Cx^d - xp$. This gives me the condition $bAx^{b-1} + dCx^{d-1}= p$. Is it possible to express the $x$ which satisfies this constraint as a function of $p$?

Answer 1

Not in a "nice" way in general. For example, the case $bA=dC=1,\,b=6,\,d=2$ doesn't allow radical solutions. Instead you'd work with a generalisation called Bring radicals, but that just names our ignorance. Your best bet is numerical methods such as Newton-Raphson.