4.18.21 $ax+x{y}^{\prime }(x{)}^2-2y(x)y^{\prime }(x)=0$

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

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

Book solution method
Homogeneous ODE, $x^{n}f(\frac{y}{x},y^{\prime })=0$, Solve for $y$

Mathematica ✓
cpu = 0.072401 (sec), leaf count = 151

$$\{\{y(x)\to -\frac{\sqrt{a}x\tan (c_1-i\log (x))}{\sqrt{{\sec }^2(c_1-i\log (x))}}\},\{y(x)\to \frac{\sqrt{a}x\tan (c_1-i\log (x))}{\sqrt{{\sec }^2(c_1-i\log (x))}}\},\{y(x)\to -\frac{\sqrt{a}x\tan (c_1+i\log (x))}{\sqrt{{\sec }^2(c_1+i\log (x))}}\},\{y(x)\to \frac{\sqrt{a}x\tan (c_1+i\log (x))}{\sqrt{{\sec }^2(c_1+i\log (x))}}\}\}$$

Maple ✓
cpu = 0.023 (sec), leaf count = 32

$$\{{\left(y\left(x\right)\right)}^2-a{x}^2=0,[x\left(\mathit{\_}\mathit{T}\right)=\mathit{\_}\mathit{T}\mathit{\_}\mathit{C}\mathit{1},y\left(\mathit{\_}\mathit{T}\right)=\frac{({\mathit{\_}\mathit{T}}^2+a)\mathit{\_}\mathit{C}\mathit{1}}{2}]\}$$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -((Sqrt[a]*x*Tan[C[1] - I*Log[x]])/Sqrt[Sec[C[1] - I*Log[x]]^2])}, {y[
x] -> (Sqrt[a]*x*Tan[C[1] - I*Log[x]])/Sqrt[Sec[C[1] - I*Log[x]]^2]}, {y[x] -> -
((Sqrt[a]*x*Tan[C[1] + I*Log[x]])/Sqrt[Sec[C[1] + I*Log[x]]^2])}, {y[x] -> (Sqrt
[a]*x*Tan[C[1] + I*Log[x]])/Sqrt[Sec[C[1] + I*Log[x]]^2]}}

Maple raw input

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

Maple raw output

y(x)^2-a*x^2 = 0, [x(_T) = _T*_C1, y(_T) = 1/2*(_T^2+a)*_C1]