4.19.15 \(x^2 y'(x)^2+(y(x)+2 x) y(x) y'(x)+y(x)^2=0\)

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

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

Book solution method
Change of variable

Mathematica
cpu = 0.131129 (sec), leaf count = 64

\[\left \{\left \{y(x)\to -\frac {\sinh \left (4 c_1\right )+\cosh \left (4 c_1\right )}{\sinh \left (2 c_1\right )+\cosh \left (2 c_1\right )-x}\right \},\left \{y(x)\to \frac {\sinh \left (4 c_1\right )+\cosh \left (4 c_1\right )}{\sinh \left (2 c_1\right )+\cosh \left (2 c_1\right )+x}\right \}\right \}\]

Maple
cpu = 0.08 (sec), leaf count = 99

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

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

Mathematica raw output

{{y[x] -> -((Cosh[4*C[1]] + Sinh[4*C[1]])/(-x + Cosh[2*C[1]] + Sinh[2*C[1]]))}, 
{y[x] -> (Cosh[4*C[1]] + Sinh[4*C[1]])/(x + Cosh[2*C[1]] + 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(((y(x)*(y(x)+4*x)/x^2)^(1/2)*x+2*x+y(x))/x)+1/2*ln(y(x
)/x)-_C1 = 0, ln(x)+1/2*ln(((y(x)*(y(x)+4*x)/x^2)^(1/2)*x+2*x+y(x))/x)+1/2*ln(y(
x)/x)-_C1 = 0