ODE
\[ x^2 (x-y(x)) y''(x)=\left (x y'(x)-y(x)\right )^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.236102 (sec), leaf count = 56
\[\left \{\left \{y(x)\to x-\sqrt {x} \sqrt {x+2 c_2 x-2 c_1}\right \},\left \{y(x)\to x+\sqrt {x} \sqrt {x+2 c_2 x-2 c_1}\right \}\right \}\]
Maple ✓
cpu = 0.172 (sec), leaf count = 45
\[\left [y \left (x \right ) = x -\sqrt {-2 x^{2} \textit {\_C1} -2 \textit {\_C2} x +x^{2}}, y \left (x \right ) = x +\sqrt {-2 x^{2} \textit {\_C1} -2 \textit {\_C2} x +x^{2}}\right ]\] Mathematica raw input
DSolve[x^2*(x - y[x])*y''[x] == (-y[x] + x*y'[x])^2,y[x],x]
Mathematica raw output
{{y[x] -> x - Sqrt[x]*Sqrt[x - 2*C[1] + 2*x*C[2]]}, {y[x] -> x + Sqrt[x]*Sqrt[x - 2*C[1] + 2*x*C[2]]}}
Maple raw input
dsolve(x^2*(x-y(x))*diff(diff(y(x),x),x) = (x*diff(y(x),x)-y(x))^2, y(x))
Maple raw output
[y(x) = x-(-2*_C1*x^2-2*_C2*x+x^2)^(1/2), y(x) = x+(-2*_C1*x^2-2*_C2*x+x^2)^(1/2)]