##### 4.17.21 $$a y(x) y'(x)-a x+y'(x)^2=0$$

ODE
$a y(x) y'(x)-a x+y'(x)^2=0$ ODE Classiﬁcation

[_dAlembert]

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica
cpu = 1.6871 (sec), leaf count = 51

$\text {Solve}\left [\left \{x=\frac {K[1] \left (\sin ^{-1}(K[1])+a c_1\right )}{a \sqrt {1-K[1]^2}},\frac {x}{K[1]}=\frac {K[1]}{a}+y(x)\right \},\{y(x),K[1]\}\right ]$

Maple
cpu = 0.672 (sec), leaf count = 394

$\left [\frac {\left (-a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) \textit {\_C1}}{\sqrt {-2 a y \left (x \right )+2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+4}\, \sqrt {-2 a y \left (x \right )+2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}-4}}+x +\frac {\left (-a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) \ln \left (-\frac {a y \left (x \right )}{2}+\frac {\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}}{2}+\frac {\sqrt {2 a^{2} y \left (x \right )^{2}-2 a y \left (x \right ) \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+4 a x -4}}{2}\right )}{a \sqrt {2 a^{2} y \left (x \right )^{2}-2 a y \left (x \right ) \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+4 a x -4}} = 0, \frac {\left (a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) \textit {\_C1}}{\sqrt {-2 a y \left (x \right )-2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+4}\, \sqrt {-2 a y \left (x \right )-2 \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}-4}}+x -\frac {\left (a y \left (x \right )+\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}\right ) \ln \left (-\frac {a y \left (x \right )}{2}-\frac {\sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}}{2}+\frac {\sqrt {2 a^{2} y \left (x \right )^{2}+2 a y \left (x \right ) \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+4 a x -4}}{2}\right )}{a \sqrt {2 a^{2} y \left (x \right )^{2}+2 a y \left (x \right ) \sqrt {a \left (a y \left (x \right )^{2}+4 x \right )}+4 a x -4}} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

Solve[{x == ((ArcSin[K[1]] + a*C[1])*K[1])/(a*Sqrt[1 - K[1]^2]), x/K[1] == K[1]/
a + y[x]}, {y[x], K[1]}]

Maple raw input

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

Maple raw output

[(-a*y(x)+(a*(a*y(x)^2+4*x))^(1/2))/(-2*a*y(x)+2*(a*(a*y(x)^2+4*x))^(1/2)+4)^(1/
2)/(-2*a*y(x)+2*(a*(a*y(x)^2+4*x))^(1/2)-4)^(1/2)*_C1+x+(-a*y(x)+(a*(a*y(x)^2+4*
x))^(1/2))*ln(-1/2*a*y(x)+1/2*(a*(a*y(x)^2+4*x))^(1/2)+1/2*(2*a^2*y(x)^2-2*a*y(x
)*(a*(a*y(x)^2+4*x))^(1/2)+4*a*x-4)^(1/2))/a/(2*a^2*y(x)^2-2*a*y(x)*(a*(a*y(x)^2
+4*x))^(1/2)+4*a*x-4)^(1/2) = 0, (a*y(x)+(a*(a*y(x)^2+4*x))^(1/2))/(-2*a*y(x)-2*
(a*(a*y(x)^2+4*x))^(1/2)+4)^(1/2)/(-2*a*y(x)-2*(a*(a*y(x)^2+4*x))^(1/2)-4)^(1/2)
*_C1+x-(a*y(x)+(a*(a*y(x)^2+4*x))^(1/2))*ln(-1/2*a*y(x)-1/2*(a*(a*y(x)^2+4*x))^(
1/2)+1/2*(2*a^2*y(x)^2+2*a*y(x)*(a*(a*y(x)^2+4*x))^(1/2)+4*a*x-4)^(1/2))/a/(2*a^
2*y(x)^2+2*a*y(x)*(a*(a*y(x)^2+4*x))^(1/2)+4*a*x-4)^(1/2) = 0]