A question on a multivariable continuously differentiable function

Question

Assume $f(x_{1},x_{2})$ is a real-valued continuously differentiable function, and assume it holds that

$x_2D_{1}f(x_1,x_2) - x_1D_2f(x_1,x_2) = 0$ where $D_1$ is the partial derivative with respect to $x_1$, similar for $D_2$.

Show there exists a function $\phi:\mathbb{R} \to \mathbb{R}$, $\phi$ continuously differentiable, such that $f(x_1,x_2) = \phi(x_1^{2} + x_2^2)$.

So far I have thought that assuming $\phi$ would exist it must satisfy $2x_1x_2D_1\phi - 2x_1x_2D_2\phi = 0$ and thus this leads to $D_1\phi = D_2\phi$, however I'm not sure if we can assume that $\phi$ exists as that is what the question asks us to prove... really not sure on how to go about it at all!

Answer 1

HINT: Show that the gradient of $f$ points radially outward (from the origin). What does this tell you about $f$?

Answer 2

Let me write it as $$xF_y-yF_x=0$$ then $$\dfrac{-F_x}{F_y}=\dfrac {-x}{y} $$ remember that $\dfrac{dy}{dx}=\dfrac{-F_x}{F_y}$ when $F(x,y)=c$ so

$$\dfrac{dy}{dx}=\dfrac {-x}{y}$$ $$-xdx=ydy\implies -\int x dx=\int ydy \implies -x^2=y^2+C$$ $$x^2+y^2=C$$

Thus, we see that $F(x,y)=c\implies x^2+y^2=C \implies F(x,y)=\phi(x^2+y^2)$

Note: I used the fact that when $F(x,y)=c$ then by implicit differantation we have $\dfrac{dy}{dx}=\dfrac{-F_x}{F_y}$.