4.18.12 $ay(x)+x{y}^{\prime }(x{)}^2-y(x)y^{\prime }(x)=0$

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

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

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

Mathematica ✓
cpu = 0.805737 (sec), leaf count = 158

$$\{\text{Solve}[\frac{y(x)}{ax}+\frac{\sqrt{\frac{y(x)}{x}}\sqrt{\frac{y(x)}{x}-4a}}{a}+4c_1+2\log (x)=4\log (\sqrt{\frac{y(x)}{x}-4a}+\sqrt{\frac{y(x)}{x}}),y(x)],\text{Solve}[\frac{\sqrt{\frac{y(x)}{x}}\sqrt{\frac{y(x)}{x}-4a}}{a}+4c_1=\frac{y(x)}{ax}+4\log (\sqrt{\frac{y(x)}{x}-4a}+\sqrt{\frac{y(x)}{x}})+2\log (x),y(x)]\}$$

Maple ✓
cpu = 0.027 (sec), leaf count = 40

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

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

Mathematica raw output

{Solve[4*C[1] + 2*Log[x] + y[x]/(a*x) + (Sqrt[y[x]/x]*Sqrt[-4*a + y[x]/x])/a == 
4*Log[Sqrt[y[x]/x] + Sqrt[-4*a + y[x]/x]], y[x]], Solve[4*C[1] + (Sqrt[y[x]/x]*S
qrt[-4*a + y[x]/x])/a == 2*Log[x] + 4*Log[Sqrt[y[x]/x] + Sqrt[-4*a + y[x]/x]] + 
y[x]/(a*x), y[x]]}

Maple raw input

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

Maple raw output

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