4.15.35 $$x \left (\sqrt {x^2+y(x)^2}+x\right ) y'(x)+\sqrt {x^2+y(x)^2} y(x)=0$$

ODE
$x \left (\sqrt {x^2+y(x)^2}+x\right ) y'(x)+\sqrt {x^2+y(x)^2} y(x)=0$ ODE Classiﬁcation

[[_homogeneous, class G], _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.537569 (sec), leaf count = 1457

$\left \{\left \{y(x)\to -\frac {1}{2} \sqrt {\frac {x^6-\sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}} x^4+\left (\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}\right ){}^{2/3} x^2+8 e^{6 c_1}}{x^2 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}}}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {\frac {x^6-\sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}} x^4+\left (\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}\right ){}^{2/3} x^2+8 e^{6 c_1}}{x^2 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}}}}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {i \left (\left (i+\sqrt {3}\right ) x^6+2 i \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}} x^4-\left (-i+\sqrt {3}\right ) \left (\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}\right ){}^{2/3} x^2+8 \left (i+\sqrt {3}\right ) e^{6 c_1}\right )}{x^2 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}}}}}{2 \sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {i \left (\left (i+\sqrt {3}\right ) x^6+2 i \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}} x^4-\left (-i+\sqrt {3}\right ) \left (\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}\right ){}^{2/3} x^2+8 \left (i+\sqrt {3}\right ) e^{6 c_1}\right )}{x^2 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}}}}}{2 \sqrt {2}}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {i \left (x^2 \left (x^2+\sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}}\right ) \left (\left (i+\sqrt {3}\right ) \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}}-\left (-i+\sqrt {3}\right ) x^2\right )-8 \left (-i+\sqrt {3}\right ) e^{6 c_1}\right )}{x^2 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}}}}}{2 \sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {\frac {i \left (x^2 \left (x^2+\sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}}\right ) \left (\left (i+\sqrt {3}\right ) \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}}-\left (-i+\sqrt {3}\right ) x^2\right )-8 \left (-i+\sqrt {3}\right ) e^{6 c_1}\right )}{x^2 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 e^{12 c_1}+8 \sqrt {e^{6 c_1} \left (e^{6 c_1}-x^6\right ){}^3}}{x^6}}}}}{2 \sqrt {2}}\right \}\right \}$

Maple
cpu = 0.129 (sec), leaf count = 139

$\left [\int _{\textit {\_b}}^{x}-\frac {\sqrt {\textit {\_a}^{2}+y \left (x \right )^{2}}}{\textit {\_a} \left (2 \sqrt {\textit {\_a}^{2}+y \left (x \right )^{2}}+\textit {\_a} \right )}d \textit {\_a} +\int _{}^{y \left (x \right )}\left (-\frac {x +\sqrt {\textit {\_f}^{2}+x^{2}}}{\textit {\_f} \left (2 \sqrt {\textit {\_f}^{2}+x^{2}}+x \right )}-\left (\int _{\textit {\_b}}^{x}\left (-\frac {\textit {\_f}}{\sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \textit {\_a} \left (2 \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}+\textit {\_a} \right )}+\frac {2 \textit {\_f}}{\textit {\_a} \left (2 \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}+\textit {\_a} \right )^{2}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +\textit {\_C1} = 0\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -1/2*Sqrt[(8*E^(6*C[1]) + x^6 - x^4*((8*E^(12*C[1]) + 20*E^(6*C[1])*x^
6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/x^6)^(1/3) + x^2*((8*E^(12*C
[1]) + 20*E^(6*C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/x^6)^
(2/3))/(x^2*((8*E^(12*C[1]) + 20*E^(6*C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6
*C[1]) - x^6)^3])/x^6)^(1/3))]}, {y[x] -> Sqrt[(8*E^(6*C[1]) + x^6 - x^4*((8*E^(
12*C[1]) + 20*E^(6*C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/x
^6)^(1/3) + x^2*((8*E^(12*C[1]) + 20*E^(6*C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(
E^(6*C[1]) - x^6)^3])/x^6)^(2/3))/(x^2*((8*E^(12*C[1]) + 20*E^(6*C[1])*x^6 - x^1
2 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/x^6)^(1/3))]/2}, {y[x] -> -1/2*Sqrt
[(I*(8*(I + Sqrt[3])*E^(6*C[1]) + (I + Sqrt[3])*x^6 + (2*I)*x^4*((8*E^(12*C[1])
+ 20*E^(6*C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/x^6)^(1/3)
 - (-I + Sqrt[3])*x^2*((8*E^(12*C[1]) + 20*E^(6*C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C
[1])*(E^(6*C[1]) - x^6)^3])/x^6)^(2/3)))/(x^2*((8*E^(12*C[1]) + 20*E^(6*C[1])*x^
6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/x^6)^(1/3))]/Sqrt[2]}, {y[x]
 -> Sqrt[(I*(8*(I + Sqrt[3])*E^(6*C[1]) + (I + Sqrt[3])*x^6 + (2*I)*x^4*((8*E^(1
2*C[1]) + 20*E^(6*C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/x^
6)^(1/3) - (-I + Sqrt[3])*x^2*((8*E^(12*C[1]) + 20*E^(6*C[1])*x^6 - x^12 + 8*Sqr
t[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/x^6)^(2/3)))/(x^2*((8*E^(12*C[1]) + 20*E^(6*
C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/x^6)^(1/3))]/(2*Sqrt
[2])}, {y[x] -> -1/2*Sqrt[(I*(-8*(-I + Sqrt[3])*E^(6*C[1]) + x^2*(x^2 + ((8*E^(1
2*C[1]) + 20*E^(6*C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/x^
6)^(1/3))*(-((-I + Sqrt[3])*x^2) + (I + Sqrt[3])*((8*E^(12*C[1]) + 20*E^(6*C[1])
*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/x^6)^(1/3))))/(x^2*((8*E^
(12*C[1]) + 20*E^(6*C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/
x^6)^(1/3))]/Sqrt[2]}, {y[x] -> Sqrt[(I*(-8*(-I + Sqrt[3])*E^(6*C[1]) + x^2*(x^2
 + ((8*E^(12*C[1]) + 20*E^(6*C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) -
x^6)^3])/x^6)^(1/3))*(-((-I + Sqrt[3])*x^2) + (I + Sqrt[3])*((8*E^(12*C[1]) + 20
*E^(6*C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1]) - x^6)^3])/x^6)^(1/3))))/
(x^2*((8*E^(12*C[1]) + 20*E^(6*C[1])*x^6 - x^12 + 8*Sqrt[E^(6*C[1])*(E^(6*C[1])
- x^6)^3])/x^6)^(1/3))]/(2*Sqrt[2])}}

Maple raw input

dsolve(x*(x+(x^2+y(x)^2)^(1/2))*diff(y(x),x)+y(x)*(x^2+y(x)^2)^(1/2) = 0, y(x))

Maple raw output

[Int(-(_a^2+y(x)^2)^(1/2)/_a/(2*(_a^2+y(x)^2)^(1/2)+_a),_a = _b .. x)+Intat(-(x+
(_f^2+x^2)^(1/2))/_f/(2*(_f^2+x^2)^(1/2)+x)-Int(-1/(_a^2+_f^2)^(1/2)/_a/(2*(_a^2
+_f^2)^(1/2)+_a)*_f+2/_a/(2*(_a^2+_f^2)^(1/2)+_a)^2*_f,_a = _b .. x),_f = y(x))+
_C1 = 0]