4.19.6 $x^2{y}^{\prime }(x{)}^2-x(x-2y(x))y^{\prime }(x)+y(x{)}^2=0$

ODE
$$x^2{y}^{\prime }(x{)}^2-x(x-2y(x))y^{\prime }(x)+y(x{)}^2=0$$ ODE Classification

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

Book solution method
Change of variable

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

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

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

$$\{\ln \left(x\right)-\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\sqrt{\frac{-4y\left(x\right)+x}{x}}\right)+\frac{1}{2}\ln \left(\frac{y\left(x\right)}{x}\right)-\mathit{\_}\mathit{C}\mathit{1}=0,\ln \left(x\right)+\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\sqrt{\frac{-4y\left(x\right)+x}{x}}\right)+\frac{1}{2}\ln \left(\frac{y\left(x\right)}{x}\right)-\mathit{\_}\mathit{C}\mathit{1}=0,y\left(x\right)=\frac{x}{4}\}$$ 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