4.19.16 \(x^2 y'(x)^2+(2 x-y(x)) y(x) y'(x)+y(x)^2=0\)

ODE
\[ x^2 y'(x)^2+(2 x-y(x)) y(x) y'(x)+y(x)^2=0 \] ODE Classification

[[_homogeneous, `class A`], _dAlembert]

Book solution method
No Missing Variables ODE, Solve for \(x\)

Mathematica
cpu = 0.132102 (sec), leaf count = 73

\[\left \{\left \{y(x)\to \frac {\sinh \left (2 c_1\right )-\cosh \left (2 c_1\right )}{x \sinh \left (2 c_1\right )+x \cosh \left (2 c_1\right )-1}\right \},\left \{y(x)\to \frac {\sinh \left (2 c_1\right )-\cosh \left (2 c_1\right )}{x \sinh \left (2 c_1\right )+x \cosh \left (2 c_1\right )+1}\right \}\right \}\]

Maple
cpu = 0.08 (sec), leaf count = 103

\[ \left \{ \ln \relax (x ) -{\frac {1}{2}\ln \left ({\frac {1}{x} \left (\sqrt {{\frac {-4\,xy \relax (x ) + \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}}}x-2\,x+y \relax (x ) \right ) } \right ) }+{\frac {1}{2}\ln \left ({\frac {y \relax (x ) }{x}} \right ) }-{\it \_C1}=0,\ln \relax (x ) +{\frac {1}{2}\ln \left ({\frac {1}{x} \left (\sqrt {{\frac {-4\,xy \relax (x ) + \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}}}x-2\,x+y \relax (x ) \right ) } \right ) }+{\frac {1}{2}\ln \left ({\frac {y \relax (x ) }{x}} \right ) }-{\it \_C1}=0,y \relax (x ) =4\,x \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-Cosh[2*C[1]] + Sinh[2*C[1]])/(-1 + x*Cosh[2*C[1]] + x*Sinh[2*C[1]])}
, {y[x] -> (-Cosh[2*C[1]] + Sinh[2*C[1]])/(1 + x*Cosh[2*C[1]] + x*Sinh[2*C[1]])}
}

Maple raw input

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

Maple raw output

y(x) = 4*x, ln(x)-1/2*ln((((-4*x*y(x)+y(x)^2)/x^2)^(1/2)*x-2*x+y(x))/x)+1/2*ln(y
(x)/x)-_C1 = 0, ln(x)+1/2*ln((((-4*x*y(x)+y(x)^2)/x^2)^(1/2)*x-2*x+y(x))/x)+1/2*
ln(y(x)/x)-_C1 = 0