##### 4.17.22 $$-a y(x) y'(x)-a x+y'(x)^2=0$$

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 = 1.20744 (sec), leaf count = 49

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

Maple
cpu = 0.229 (sec), leaf count = 183

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

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

Mathematica raw output

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

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