4.18.8 \(a+x y'(x)^2-y(x) y'(x)=0\)

ODE
\[ a+x y'(x)^2-y(x) y'(x)=0 \] 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

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

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

\[ \left \{ \left (y \relax (x ) \right ) ^{2}-4\,ax=0,y \relax (x ) ={\frac {x{{\it \_C1}}^{2}+a}{{\it \_C1}}} \right \} \] 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