4.15.35 $x(\sqrt{x^2+y(x{)}^2}+x)y^{\prime }(x)+\sqrt{x^2+y(x{)}^2}y(x)=0$

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

[[_homogeneous, `class G`], _dAlembert]

Book solution method
Homogeneous equation

Mathematica ✓
cpu = 1.15496 (sec), leaf count = 2949

$$\{\{y(x)\to -\frac{1}{2}\sqrt{\frac{x^{16}+8e^{6c_1}{x}^{10}-\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}}{x}^8+(-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}){}^{2/3}}{x^6\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}}}}\},\{y(x)\to \frac{1}{2}\sqrt{\frac{x^{16}+8e^{6c_1}{x}^{10}-\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}}{x}^8+(-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}){}^{2/3}}{x^6\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}}}}\},\{y(x)\to -\frac{\sqrt{-\frac{i((x^8+\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}})((-i+\sqrt{3})\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}}-(i+\sqrt{3})x^8)-8(i+\sqrt{3})e^{6c_1}{x}^{10})}{x^6\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}}}}}{2\sqrt{2}}\},\{y(x)\to \frac{\sqrt{-\frac{i((x^8+\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}})((-i+\sqrt{3})\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}}-(i+\sqrt{3})x^8)-8(i+\sqrt{3})e^{6c_1}{x}^{10})}{x^6\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}}}}}{2\sqrt{2}}\},\{y(x)\to -\frac{\sqrt{\frac{i((x^8+\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}})((i+\sqrt{3})\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}}-(-i+\sqrt{3})x^8)-8(-i+\sqrt{3})e^{6c_1}{x}^{10})}{x^6\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}}}}}{2\sqrt{2}}\},\{y(x)\to \frac{\sqrt{\frac{i((x^8+\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}})((i+\sqrt{3})\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}}-(-i+\sqrt{3})x^8)-8(-i+\sqrt{3})e^{6c_1}{x}^{10})}{x^6\sqrt[3]{-x^{24}+20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(e^{6c_1}-x^6){}^3}}}}}{2\sqrt{2}}\},\{y(x)\to -\frac{1}{2}\sqrt{\frac{x^{16}-8e^{6c_1}{x}^{10}-\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}}{x}^8+(-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}){}^{2/3}}{x^6\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}}}}\},\{y(x)\to \frac{1}{2}\sqrt{\frac{x^{16}-8e^{6c_1}{x}^{10}-\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}}{x}^8+(-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}){}^{2/3}}{x^6\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}}}}\},\{y(x)\to -\frac{\sqrt{\frac{8(1-i\sqrt{3})e^{6c_1}{x}^{10}-i(x^8+\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}})((-i+\sqrt{3})\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}}-(i+\sqrt{3})x^8)}{x^6\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}}}}}{2\sqrt{2}}\},\{y(x)\to \frac{\sqrt{\frac{8(1-i\sqrt{3})e^{6c_1}{x}^{10}-i(x^8+\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}})((-i+\sqrt{3})\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}}-(i+\sqrt{3})x^8)}{x^6\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}}}}}{2\sqrt{2}}\},\{y(x)\to -\frac{\sqrt{\frac{8(1+i\sqrt{3})e^{6c_1}{x}^{10}+i(x^8+\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}})((i+\sqrt{3})\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}}-(-i+\sqrt{3})x^8)}{x^6\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}}}}}{2\sqrt{2}}\},\{y(x)\to \frac{\sqrt{\frac{8(1+i\sqrt{3})e^{6c_1}{x}^{10}+i(x^8+\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}})((i+\sqrt{3})\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}}-(-i+\sqrt{3})x^8)}{x^6\sqrt[3]{-x^{24}-20e^{6c_1}{x}^{18}+8e^{12c_1}{x}^{12}+8\sqrt{e^{6c_1}{x}^{24}(x^6+e^{6c_1}){}^3}}}}}{2\sqrt{2}}\}\}$$

Maple ✓
cpu = 0.132 (sec), leaf count = 133

$$\{{\int }_{\mathit{\_}\mathit{b}}^x-\frac{1}{\mathit{\_}\mathit{a}}\sqrt{{\mathit{\_}\mathit{a}}^2+{\left(y\left(x\right)\right)}^2}{(2\sqrt{{\mathit{\_}\mathit{a}}^2+{\left(y\left(x\right)\right)}^2}+\mathit{\_}\mathit{a})}^{-1}\mathrm{d}\mathit{\_}\mathit{a}+{\int }^{y\left(x\right)}-\frac{1}{\mathit{\_}\mathit{f}}(x+\sqrt{{\mathit{\_}\mathit{f}}^2+x^2}){(2\sqrt{{\mathit{\_}\mathit{f}}^2+x^2}+x)}^{-1}-{\int }_{\mathit{\_}\mathit{b}}^x-\frac{\mathit{\_}\mathit{f}}{\mathit{\_}\mathit{a}}(-2+1(2\sqrt{{\mathit{\_}\mathit{a}}^2+{\mathit{\_}\mathit{f}}^2}+\mathit{\_}\mathit{a})\frac{1}{\sqrt{{\mathit{\_}\mathit{a}}^2+{\mathit{\_}\mathit{f}}^2}}){(2\sqrt{{\mathit{\_}\mathit{a}}^2+{\mathit{\_}\mathit{f}}^2}+\mathit{\_}\mathit{a})}^{-2}\mathrm{d}\mathit{\_}\mathit{a}d\mathit{\_}\mathit{f}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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] -> -Sqrt[(8*E^(6*C[1])*x^10 + x^16 - x^8*(8*E^(12*C[1])*x^12 + 20*E^(6*C[
1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) - x^6)^3])^(1/3) + (8*E^(12
*C[1])*x^12 + 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) - x
^6)^3])^(2/3))/(x^6*(8*E^(12*C[1])*x^12 + 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(
6*C[1])*x^24*(E^(6*C[1]) - x^6)^3])^(1/3))]/2}, {y[x] -> Sqrt[(8*E^(6*C[1])*x^10
 + x^16 - x^8*(8*E^(12*C[1])*x^12 + 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1]
)*x^24*(E^(6*C[1]) - x^6)^3])^(1/3) + (8*E^(12*C[1])*x^12 + 20*E^(6*C[1])*x^18 -
 x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) - x^6)^3])^(2/3))/(x^6*(8*E^(12*C[1])
*x^12 + 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) - x^6)^3]
)^(1/3))]/2}, {y[x] -> -Sqrt[((-I)*(-8*(I + Sqrt[3])*E^(6*C[1])*x^10 + (x^8 + (8
*E^(12*C[1])*x^12 + 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1
]) - x^6)^3])^(1/3))*(-((I + Sqrt[3])*x^8) + (-I + Sqrt[3])*(8*E^(12*C[1])*x^12 
+ 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) - x^6)^3])^(1/3
))))/(x^6*(8*E^(12*C[1])*x^12 + 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^
24*(E^(6*C[1]) - x^6)^3])^(1/3))]/(2*Sqrt[2])}, {y[x] -> Sqrt[((-I)*(-8*(I + Sqr
t[3])*E^(6*C[1])*x^10 + (x^8 + (8*E^(12*C[1])*x^12 + 20*E^(6*C[1])*x^18 - x^24 +
 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) - x^6)^3])^(1/3))*(-((I + Sqrt[3])*x^8) + (-
I + Sqrt[3])*(8*E^(12*C[1])*x^12 + 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])
*x^24*(E^(6*C[1]) - x^6)^3])^(1/3))))/(x^6*(8*E^(12*C[1])*x^12 + 20*E^(6*C[1])*x
^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) - x^6)^3])^(1/3))]/(2*Sqrt[2])},
 {y[x] -> -Sqrt[(I*(-8*(-I + Sqrt[3])*E^(6*C[1])*x^10 + (x^8 + (8*E^(12*C[1])*x^
12 + 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) - x^6)^3])^(
1/3))*(-((-I + Sqrt[3])*x^8) + (I + Sqrt[3])*(8*E^(12*C[1])*x^12 + 20*E^(6*C[1])
*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) - x^6)^3])^(1/3))))/(x^6*(8*E^
(12*C[1])*x^12 + 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) 
- x^6)^3])^(1/3))]/(2*Sqrt[2])}, {y[x] -> Sqrt[(I*(-8*(-I + Sqrt[3])*E^(6*C[1])*
x^10 + (x^8 + (8*E^(12*C[1])*x^12 + 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1]
)*x^24*(E^(6*C[1]) - x^6)^3])^(1/3))*(-((-I + Sqrt[3])*x^8) + (I + Sqrt[3])*(8*E
^(12*C[1])*x^12 + 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1])
 - x^6)^3])^(1/3))))/(x^6*(8*E^(12*C[1])*x^12 + 20*E^(6*C[1])*x^18 - x^24 + 8*Sq
rt[E^(6*C[1])*x^24*(E^(6*C[1]) - x^6)^3])^(1/3))]/(2*Sqrt[2])}, {y[x] -> -Sqrt[(
-8*E^(6*C[1])*x^10 + x^16 - x^8*(8*E^(12*C[1])*x^12 - 20*E^(6*C[1])*x^18 - x^24 
+ 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) + x^6)^3])^(1/3) + (8*E^(12*C[1])*x^12 - 20
*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) + x^6)^3])^(2/3))/(
x^6*(8*E^(12*C[1])*x^12 - 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^
(6*C[1]) + x^6)^3])^(1/3))]/2}, {y[x] -> Sqrt[(-8*E^(6*C[1])*x^10 + x^16 - x^8*(
8*E^(12*C[1])*x^12 - 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[
1]) + x^6)^3])^(1/3) + (8*E^(12*C[1])*x^12 - 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[
E^(6*C[1])*x^24*(E^(6*C[1]) + x^6)^3])^(2/3))/(x^6*(8*E^(12*C[1])*x^12 - 20*E^(6
*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) + x^6)^3])^(1/3))]/2}, {
y[x] -> -Sqrt[(8*(1 - I*Sqrt[3])*E^(6*C[1])*x^10 - I*(x^8 + (8*E^(12*C[1])*x^12 
- 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) + x^6)^3])^(1/3
))*(-((I + Sqrt[3])*x^8) + (-I + Sqrt[3])*(8*E^(12*C[1])*x^12 - 20*E^(6*C[1])*x^
18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) + x^6)^3])^(1/3)))/(x^6*(8*E^(12*
C[1])*x^12 - 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) + x^
6)^3])^(1/3))]/(2*Sqrt[2])}, {y[x] -> Sqrt[(8*(1 - I*Sqrt[3])*E^(6*C[1])*x^10 - 
I*(x^8 + (8*E^(12*C[1])*x^12 - 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^2
4*(E^(6*C[1]) + x^6)^3])^(1/3))*(-((I + Sqrt[3])*x^8) + (-I + Sqrt[3])*(8*E^(12*
C[1])*x^12 - 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) + x^
6)^3])^(1/3)))/(x^6*(8*E^(12*C[1])*x^12 - 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(
6*C[1])*x^24*(E^(6*C[1]) + x^6)^3])^(1/3))]/(2*Sqrt[2])}, {y[x] -> -Sqrt[(8*(1 +
 I*Sqrt[3])*E^(6*C[1])*x^10 + I*(x^8 + (8*E^(12*C[1])*x^12 - 20*E^(6*C[1])*x^18 
- x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) + x^6)^3])^(1/3))*(-((-I + Sqrt[3])*
x^8) + (I + Sqrt[3])*(8*E^(12*C[1])*x^12 - 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^
(6*C[1])*x^24*(E^(6*C[1]) + x^6)^3])^(1/3)))/(x^6*(8*E^(12*C[1])*x^12 - 20*E^(6*
C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) + x^6)^3])^(1/3))]/(2*Sqr
t[2])}, {y[x] -> Sqrt[(8*(1 + I*Sqrt[3])*E^(6*C[1])*x^10 + I*(x^8 + (8*E^(12*C[1
])*x^12 - 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) + x^6)^
3])^(1/3))*(-((-I + Sqrt[3])*x^8) + (I + Sqrt[3])*(8*E^(12*C[1])*x^12 - 20*E^(6*
C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[1]) + x^6)^3])^(1/3)))/(x^6*(
8*E^(12*C[1])*x^12 - 20*E^(6*C[1])*x^18 - x^24 + 8*Sqrt[E^(6*C[1])*x^24*(E^(6*C[
1]) + x^6)^3])^(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),'implicit')

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