4.19.16 $x^2{y}^{\prime }(x{)}^2+(2x-y(x))y(x)y^{\prime }(x)+y(x{)}^2=0$

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

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

Book solution method
No Missing Variables ODE, Solve for $x$

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

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

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

$$\{\ln \left(x\right)-\frac{1}{2}\ln \left(\frac{1}{x}(\sqrt{\frac{-4xy\left(x\right)+{\left(y\left(x\right)\right)}^2}{x^2}}x-2x+y\left(x\right))\right)+\frac{1}{2}\ln \left(\frac{y\left(x\right)}{x}\right)-\mathit{\_}\mathit{C}\mathit{1}=0,\ln \left(x\right)+\frac{1}{2}\ln \left(\frac{1}{x}(\sqrt{\frac{-4xy\left(x\right)+{\left(y\left(x\right)\right)}^2}{x^2}}x-2x+y\left(x\right))\right)+\frac{1}{2}\ln \left(\frac{y\left(x\right)}{x}\right)-\mathit{\_}\mathit{C}\mathit{1}=0,y\left(x\right)=4x\}$$ 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