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 = 1.15496 (sec), leaf count = 2949

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

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

\[ \left \{ \int _{{\it \_b}}^{x}\!-{\frac {1}{{\it \_a}}\sqrt {{{\it \_a}}^{2}+ \left (y \relax (x ) \right ) ^{2}} \left (2\,\sqrt {{{\it \_a}}^{2}+ \left (y \relax (x ) \right ) ^{2}}+{\it \_a} \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \relax (x ) }\!-{\frac {1}{{\it \_f}} \left (x+\sqrt {{{\it \_f}}^{2}+{x}^{2}} \right ) \left (2\,\sqrt {{{\it \_f}}^{2}+{x}^{2}}+x \right ) ^{-1}}-\int _{{\it \_b}}^{x}\!-{\frac {{\it \_f}}{{\it \_a}} \left (-2+{1 \left (2\,\sqrt {{{\it \_a}}^{2}+{{\it \_f}}^{2}}+{\it \_a} \right ) {\frac {1}{\sqrt {{{\it \_a}}^{2}+{{\it \_f}}^{2}}}}} \right ) \left (2\,\sqrt {{{\it \_a}}^{2}+{{\it \_f}}^{2}}+{\it \_a} \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0 \right \} \] 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] -> -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