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 = 0.209988 (sec), leaf count = 147

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

Maple
cpu = 0.053 (sec), leaf count = 137

\[ \left \{ \ln \relax (x ) -{\frac {y \relax (x ) }{2\,x}\sqrt {{\frac { \left (y \relax (x ) \right ) ^{2}-{x}^{2}}{{x}^{2}}}}}+{\frac {1}{2}\ln \left ({\frac {1}{x} \left (\sqrt {{\frac { \left (y \relax (x ) \right ) ^{2}-{x}^{2}}{{x}^{2}}}}x+y \relax (x ) \right ) } \right ) }+{\frac { \left (y \relax (x ) \right ) ^{2}}{2\,{x}^{2}}}-{\it \_C1}=0,\ln \relax (x ) +{\frac {y \relax (x ) }{2\,x}\sqrt {{\frac { \left (y \relax (x ) \right ) ^{2}-{x}^{2}}{{x}^{2}}}}}-{\frac {1}{2}\ln \left ({\frac {1}{x} \left (\sqrt {{\frac { \left (y \relax (x ) \right ) ^{2}-{x}^{2}}{{x}^{2}}}}x+y \relax (x ) \right ) } \right ) }+{\frac { \left (y \relax (x ) \right ) ^{2}}{2\,{x}^{2}}}-{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

ln(x)-1/2*((y(x)^2-x^2)/x^2)^(1/2)/x*y(x)+1/2*ln((((y(x)^2-x^2)/x^2)^(1/2)*x+y(x
))/x)+1/2/x^2*y(x)^2-_C1 = 0, ln(x)+1/2*((y(x)^2-x^2)/x^2)^(1/2)/x*y(x)-1/2*ln((
((y(x)^2-x^2)/x^2)^(1/2)*x+y(x))/x)+1/2/x^2*y(x)^2-_C1 = 0