4.19.15 $x^2{y}^{\prime }(x{)}^2+(y(x)+2x)y(x)y^{\prime }(x)+y(x{)}^2=0$

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

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

Book solution method
Change of variable

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

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

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

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

DSolve[y[x]^2 + 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