Solution of Nonlinear 2nd order homogeneous ODE

Question

I have the following two Nonlinear 2nd order ODE $$ x z(x) z''(x)-\bigl((d-2)x+(d-1)z(x) z'(x)\bigr)(1+z'(x)^2)=0\\ x z(x) z''(x)+\left((d-1) x+(d-2) z(x) z'(x)\right)\left(1+z'(x)^2\right)=0 $$ I have tried the methods of solving this type of eqn but it doesn't fall into any category. It explicitly depends on both x and z. It is homogeneous. Under the change $x\to a x~~\text{and}~~z\to a z$, the $\text{eqn}\to a~\text{eqn}$. This means we can express the equation in the form $F({z(x)\over x},z'(x),xz''(x))$ viz. $$ x z''(x)-\Biggl((d-2)(\frac{x}{z(x)})+(d-1)z'(x)\Biggr)(1+z'(x)^2)=0 $$ Now making the following substitution $t=ln(x)$ and $u={z\over x}$in the above equation. $$ u(u_t+u_{tt})-\Biggl((d-1)u(u+u_t)+(d-2)\Biggr)(1+(u+u_t)^2)=0 $$ Where $u_t$ denotes differentiation w.r.t t. This is an autonomous equation. Making the substitution $u_t=w(u)$ we get $$ u w(u)(1+w_u)-\Biggl((d-1)u\left(u+w(u)\right)+(d-2)\Biggr)\left(1+(u+w(u))^2\right)=0 $$ Mathematica gives the following lengthy solutions. $$ \left\{w(u)\to \frac{1}{u \left(-\frac{u^{1-d}}{\left(u^2+1\right)^{3/2} \sqrt{c_1-\frac{2 u^{4-2 d}}{2 u^2+2}}}-\frac{d u^3+d u-u^3-2 u}{u \left(d u^4+2 d u^2+d-u^4-3 u^2-2\right)}\right)}\right\},\left\{w(u)\to \frac{1}{u \left(\frac{u^{1-d}}{\left(u^2+1\right)^{3/2} \sqrt{c_1-\frac{2 u^{4-2 d}}{2 u^2+2}}}-\frac{d u^3+d u-u^3-2 u}{u \left(d u^4+2 d u^2+d-u^4-3 u^2-2\right)}\right)}\right\} $$ Mathematica is not able to solve the other equation $u_t=w(u)$ and maple is also giving some absurd results. However, I know that both of them have the solution $z(x)=\sqrt{l^2-x^2}$ with appropriate boundary conditions. Please suggest me how to proceed to obtain this result.