Find the extremals of the functional $\int\sqrt{x^2+y^2}\sqrt{1+(y'(x))^2}dx$?

Question

I want to use the polar coordinates $x=r\cos\theta$ and $y=r\sin\theta$. After transformation, I get $$\int r\sqrt{r^2+(r')^2}d\theta.$$ Then, I derived the Euler-Lagrange equation $$2r^2-rr''+3(r')^2=0.$$ I get the solutions to this ODE, but it doesn't match the solutions. $$\theta+C_1=\frac{1}{2}\arctan(\frac{\sqrt{r^4-C^2}}{C}), \text{ where } C,C_1\in\mathbb{R}.$$ Solution is $x^2\cos(\alpha)+2xy\sin(\alpha)-y^2\cos(\alpha)=\beta$, where $\alpha$, $\beta$ are constants.