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

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

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

Book solution method
Change of variable

Mathematica
cpu = 0.107745 (sec), leaf count = 85

\[\left \{\left \{y(x)\to \frac {\left (\cosh \left (4 c_1\right )-\sinh \left (4 c_1\right )\right ) \left (2 x \sinh \left (2 c_1\right )+2 x \cosh \left (2 c_1\right )-1\right )}{4 x}\right \},\left \{y(x)\to -\frac {\left (\cosh \left (4 c_1\right )-\sinh \left (4 c_1\right )\right ) \left (2 x \sinh \left (2 c_1\right )+2 x \cosh \left (2 c_1\right )+1\right )}{4 x}\right \}\right \}\]

Maple
cpu = 0.06 (sec), leaf count = 69

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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