4.18.8 $a+x{y}^{\prime }(x{)}^2-y(x)y^{\prime }(x)=0$

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

[[_homogeneous, `class G`], _rational, _Clairaut]

Book solution method
Clairaut’s equation and related types, main form

Mathematica ✓
cpu = 0.370407 (sec), leaf count = 183

$$\{\{y(x)\to -\frac{-8a^2-\sqrt{a(\sinh \left(2c_1\right)+\cosh \left(2c_1\right))((-4a+x-1)\sinh \left(\frac{c_1}{2}\right)+(4a-x-1)\cosh \left(\frac{c_1}{2}\right)){}^2}+2a\sinh \left(c_1\right)+2a\cosh \left(c_1\right)-2ax}{-4a+\sinh \left(c_1\right)+\cosh \left(c_1\right)}\},\{y(x)\to -\frac{-8a^2+\sqrt{a(\sinh \left(2c_1\right)+\cosh \left(2c_1\right))((-4a+x-1)\sinh \left(\frac{c_1}{2}\right)+(4a-x-1)\cosh \left(\frac{c_1}{2}\right)){}^2}+2a\sinh \left(c_1\right)+2a\cosh \left(c_1\right)-2ax}{-4a+\sinh \left(c_1\right)+\cosh \left(c_1\right)}\}\}$$

Maple ✓
cpu = 0.022 (sec), leaf count = 26

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

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

Mathematica raw output

{{y[x] -> -((-8*a^2 - 2*a*x + 2*a*Cosh[C[1]] + 2*a*Sinh[C[1]] - Sqrt[a*((-1 + 4*
a - x)*Cosh[C[1]/2] + (-1 - 4*a + x)*Sinh[C[1]/2])^2*(Cosh[2*C[1]] + Sinh[2*C[1]
])])/(-4*a + Cosh[C[1]] + Sinh[C[1]]))}, {y[x] -> -((-8*a^2 - 2*a*x + 2*a*Cosh[C
[1]] + 2*a*Sinh[C[1]] + Sqrt[a*((-1 + 4*a - x)*Cosh[C[1]/2] + (-1 - 4*a + x)*Sin
h[C[1]/2])^2*(Cosh[2*C[1]] + Sinh[2*C[1]])])/(-4*a + Cosh[C[1]] + Sinh[C[1]]))}}

Maple raw input

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

Maple raw output

y(x)^2-4*a*x = 0, y(x) = (_C1^2*x+a)/_C1