DSE 2015 Question 31: Multivariable calculus

For a strictly increasing differentiable $u:\mathbb{R}^2\to\mathbb{R}$, consider $\tfrac{D_1u(x_1,x_2)}{D_2u(x_1,x_2)}=\tfrac{p_1}{p_2}$ and $p_1x_1+p_2x_2=w$. What guarantees existence of continuously differentiable demand functions $x_1(p_1,p_2,w),x_2(p_1,p_2,w)$ solving these for all positive $p_1,p_2,w$?