##### 4.18.47 $$x^2 y'(x)^2+x^2-y(x)^2=0$$

ODE
$x^2 y'(x)^2+x^2-y(x)^2=0$ ODE Classiﬁcation

[[_homogeneous, class A], _rational, _dAlembert]

Book solution method
No Missing Variables ODE, Solve for $$y'$$

Mathematica
cpu = 3.35538 (sec), leaf count = 188

$\left \{\text {Solve}\left [\frac {y(x)^2-x y(x) \sqrt {\frac {y(x)+x}{x}} \sqrt {\frac {y(x)}{x}-1}}{2 x^2}+\frac {\left (\frac {y(x)}{x}-1\right ) \sin ^{-1}\left (\frac {\sqrt {1-\frac {y(x)}{x}}}{\sqrt {2}}\right )}{\sqrt {-\left (\frac {y(x)}{x}-1\right )^2}}+\log (x)=c_1,y(x)\right ],\text {Solve}\left [\frac {\left (\frac {y(x)}{x}-1\right ) \sin ^{-1}\left (\frac {\sqrt {1-\frac {y(x)}{x}}}{\sqrt {2}}\right )}{\sqrt {-\left (\frac {y(x)}{x}-1\right )^2}}=\frac {y(x) \left (x \sqrt {\frac {y(x)+x}{x}} \sqrt {\frac {y(x)}{x}-1}+y(x)\right )}{2 x^2}+\log (x)+c_1,y(x)\right ]\right \}$

Maple
cpu = 6.158 (sec), leaf count = 44

$\left [y \left (x \right ) = \frac {x \left (\LambertW \left (-\textit {\_C1} \,{\mathrm e} x^{4}\right )-1\right )}{2 \LambertW \left (-\textit {\_C1} \,{\mathrm e} x^{4}\right ) \sqrt {-\frac {1}{\LambertW \left (-\textit {\_C1} \,{\mathrm e} x^{4}\right )}}}\right ]$ Mathematica raw input

DSolve[x^2 - y[x]^2 + x^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[Log[x] + (ArcSin[Sqrt[1 - y[x]/x]/Sqrt[2]]*(-1 + y[x]/x))/Sqrt[-(-1 + y[x
]/x)^2] + (y[x]^2 - x*y[x]*Sqrt[(x + y[x])/x]*Sqrt[-1 + y[x]/x])/(2*x^2) == C[1]
, y[x]], Solve[(ArcSin[Sqrt[1 - y[x]/x]/Sqrt[2]]*(-1 + y[x]/x))/Sqrt[-(-1 + y[x]
/x)^2] == C[1] + Log[x] + (y[x]*(y[x] + x*Sqrt[(x + y[x])/x]*Sqrt[-1 + y[x]/x]))
/(2*x^2), y[x]]}

Maple raw input

dsolve(x^2*diff(y(x),x)^2+x^2-y(x)^2 = 0, y(x))

Maple raw output

[y(x) = 1/2*x*(LambertW(-_C1*exp(1)*x^4)-1)/LambertW(-_C1*exp(1)*x^4)/(-1/Lamber
tW(-_C1*exp(1)*x^4))^(1/2)]