4.4.46 \(x y'(x)+2 y(x)=\sqrt {y(x)^2+1}\)

ODE
\[ x y'(x)+2 y(x)=\sqrt {y(x)^2+1} \] ODE Classification

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 1.00548 (sec), leaf count = 3935

\[\left \{\left \{y(x)\to -\frac {\sqrt {\frac {\cosh \left (6 c_1\right )}{x^6}-\frac {\sinh \left (6 c_1\right )}{x^6}+\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^6}+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+36} x^3+\sqrt {\frac {2 \cosh \left (6 c_1\right )}{x^6}-\frac {2 \sinh \left (6 c_1\right )}{x^6}-\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^6}+\frac {36\ 2^{2/3} \left (\sinh \left (3 c_1\right )-\cosh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}-\frac {2 \left (432 \cosh \left (3 c_1\right ) x^6-432 \sinh \left (3 c_1\right ) x^6-\cosh \left (9 c_1\right )+\sinh \left (9 c_1\right )\right )}{x^9 \sqrt {\frac {\cosh \left (6 c_1\right )}{x^6}-\frac {\sinh \left (6 c_1\right )}{x^6}+\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^6}+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+36}}+72} x^3+\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )}{18 x^3}\right \},\left \{y(x)\to \frac {-\sqrt {\frac {\cosh \left (6 c_1\right )}{x^6}-\frac {\sinh \left (6 c_1\right )}{x^6}+\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^6}+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+36} x^3+\sqrt {\frac {2 \cosh \left (6 c_1\right )}{x^6}-\frac {2 \sinh \left (6 c_1\right )}{x^6}-\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^6}+\frac {36\ 2^{2/3} \left (\sinh \left (3 c_1\right )-\cosh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}-\frac {2 \left (432 \cosh \left (3 c_1\right ) x^6-432 \sinh \left (3 c_1\right ) x^6-\cosh \left (9 c_1\right )+\sinh \left (9 c_1\right )\right )}{x^9 \sqrt {\frac {\cosh \left (6 c_1\right )}{x^6}-\frac {\sinh \left (6 c_1\right )}{x^6}+\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^6}+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+36}}+72} x^3-\cosh \left (3 c_1\right )+\sinh \left (3 c_1\right )}{18 x^3}\right \},\left \{y(x)\to \frac {\sqrt {\frac {\cosh \left (6 c_1\right )}{x^6}-\frac {\sinh \left (6 c_1\right )}{x^6}+\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^6}+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+36} x^3-\sqrt {\frac {2 \cosh \left (6 c_1\right )}{x^6}-\frac {2 \sinh \left (6 c_1\right )}{x^6}-\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^6}+\frac {36\ 2^{2/3} \left (\sinh \left (3 c_1\right )-\cosh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+\frac {2 \left (432 \cosh \left (3 c_1\right ) x^6-432 \sinh \left (3 c_1\right ) x^6-\cosh \left (9 c_1\right )+\sinh \left (9 c_1\right )\right )}{x^9 \sqrt {\frac {\cosh \left (6 c_1\right )}{x^6}-\frac {\sinh \left (6 c_1\right )}{x^6}+\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^6}+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+36}}+72} x^3-\cosh \left (3 c_1\right )+\sinh \left (3 c_1\right )}{18 x^3}\right \},\left \{y(x)\to \frac {\sqrt {\frac {\cosh \left (6 c_1\right )}{x^6}-\frac {\sinh \left (6 c_1\right )}{x^6}+\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^6}+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+36} x^3+\sqrt {\frac {2 \cosh \left (6 c_1\right )}{x^6}-\frac {2 \sinh \left (6 c_1\right )}{x^6}-\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^6}+\frac {36\ 2^{2/3} \left (\sinh \left (3 c_1\right )-\cosh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+\frac {2 \left (432 \cosh \left (3 c_1\right ) x^6-432 \sinh \left (3 c_1\right ) x^6-\cosh \left (9 c_1\right )+\sinh \left (9 c_1\right )\right )}{x^9 \sqrt {\frac {\cosh \left (6 c_1\right )}{x^6}-\frac {\sinh \left (6 c_1\right )}{x^6}+\frac {9 \sqrt [3]{2} \sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}{x^6}+\frac {36\ 2^{2/3} \left (\cosh \left (3 c_1\right )-\sinh \left (3 c_1\right )\right ) \left (\left (2 x^6-1\right ) \cosh \left (3 c_1\right )+\left (2 x^6+1\right ) \sinh \left (3 c_1\right )\right )}{\sqrt [3]{32 x^{18}+40 \cosh \left (6 c_1\right ) x^{12}-40 \sinh \left (6 c_1\right ) x^{12}-\cosh \left (12 c_1\right ) x^6+\sinh \left (12 c_1\right ) x^6+\left (\cosh \left (18 c_1\right )-\sinh \left (18 c_1\right )\right ) \sqrt {x^{12} \left (\left (16 x^6+1\right ) \cosh \left (3 c_1\right )+\left (16 x^6-1\right ) \sinh \left (3 c_1\right )\right ){}^3 \left (\cosh \left (21 c_1\right )+\sinh \left (21 c_1\right )\right )}}}+36}}+72} x^3-\cosh \left (3 c_1\right )+\sinh \left (3 c_1\right )}{18 x^3}\right \}\right \}\]

Maple
cpu = 0.039 (sec), leaf count = 27

\[ \left \{ \ln \relax (x ) +\int ^{y \relax (x ) }\! \left (2\,{\it \_a}-\sqrt {{{\it \_a}}^{2}+1} \right ) ^{-1}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[2*y[x] + x*y'[x] == Sqrt[1 + y[x]^2],y[x],x]

Mathematica raw output

{{y[x] -> -(Cosh[3*C[1]] - Sinh[3*C[1]] + x^3*Sqrt[36 + Cosh[6*C[1]]/x^6 - Sinh[
6*C[1]]/x^6 + (36*2^(2/3)*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1
]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^18 + 40*x^12*Cosh[6*C[1]] - x^6*Cosh[12*C[
1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])
*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C
[1]] + Sinh[21*C[1]])])^(1/3) + (9*2^(1/3)*(32*x^18 + 40*x^12*Cosh[6*C[1]] - x^6
*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[18*C[1]] - Sin
h[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3
*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^6] + x^3*Sqrt[72 + (2*Cosh[6*C[1]])/
x^6 - (2*Sinh[6*C[1]])/x^6 + (36*2^(2/3)*(-Cosh[3*C[1]] + Sinh[3*C[1]])*((-1 + 2
*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^18 + 40*x^12*Cosh[6*C[1]] 
- x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[18*C[1]] 
- Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1
]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) - (9*2^(1/3)*(32*x^18 + 40*x^12*Co
sh[6*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cos
h[18*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)
*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^6 - (2*(432*x^6*Cosh
[3*C[1]] - Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(x^9*Sqrt[36 + C
osh[6*C[1]]/x^6 - Sinh[6*C[1]]/x^6 + (36*2^(2/3)*(Cosh[3*C[1]] - Sinh[3*C[1]])*(
(-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^18 + 40*x^12*Cosh[6
*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[18
*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sin
h[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + (9*2^(1/3)*(32*x^18 + 40*
x^12*Cosh[6*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]]
 + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 
16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^6])])/(18*x^3
)}, {y[x] -> (-Cosh[3*C[1]] + Sinh[3*C[1]] - x^3*Sqrt[36 + Cosh[6*C[1]]/x^6 - Si
nh[6*C[1]]/x^6 + (36*2^(2/3)*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh[3*
C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^18 + 40*x^12*Cosh[6*C[1]] - x^6*Cosh[12
*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1
]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[2
1*C[1]] + Sinh[21*C[1]])])^(1/3) + (9*2^(1/3)*(32*x^18 + 40*x^12*Cosh[6*C[1]] - 
x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[18*C[1]] - 
Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]]
)^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^6] + x^3*Sqrt[72 + (2*Cosh[6*C[1]
])/x^6 - (2*Sinh[6*C[1]])/x^6 + (36*2^(2/3)*(-Cosh[3*C[1]] + Sinh[3*C[1]])*((-1 
+ 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^18 + 40*x^12*Cosh[6*C[1
]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[18*C[1
]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*
C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) - (9*2^(1/3)*(32*x^18 + 40*x^12
*Cosh[6*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (
Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x
^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^6 - (2*(432*x^6*C
osh[3*C[1]] - Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(x^9*Sqrt[36 
+ Cosh[6*C[1]]/x^6 - Sinh[6*C[1]]/x^6 + (36*2^(2/3)*(Cosh[3*C[1]] - Sinh[3*C[1]]
)*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^18 + 40*x^12*Cos
h[6*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh
[18*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*
Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + (9*2^(1/3)*(32*x^18 + 
40*x^12*Cosh[6*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[
1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1
 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^6])])/(18*
x^3)}, {y[x] -> (-Cosh[3*C[1]] + Sinh[3*C[1]] + x^3*Sqrt[36 + Cosh[6*C[1]]/x^6 -
 Sinh[6*C[1]]/x^6 + (36*2^(2/3)*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*Cosh
[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^18 + 40*x^12*Cosh[6*C[1]] - x^6*Cosh
[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*
C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cos
h[21*C[1]] + Sinh[21*C[1]])])^(1/3) + (9*2^(1/3)*(32*x^18 + 40*x^12*Cosh[6*C[1]]
 - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[18*C[1]]
 - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[
1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^6] - x^3*Sqrt[72 + (2*Cosh[6*C
[1]])/x^6 - (2*Sinh[6*C[1]])/x^6 + (36*2^(2/3)*(-Cosh[3*C[1]] + Sinh[3*C[1]])*((
-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^18 + 40*x^12*Cosh[6*
C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[18*
C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh
[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) - (9*2^(1/3)*(32*x^18 + 40*x
^12*Cosh[6*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] 
+ (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 1
6*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^6 + (2*(432*x^
6*Cosh[3*C[1]] - Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(x^9*Sqrt[
36 + Cosh[6*C[1]]/x^6 - Sinh[6*C[1]]/x^6 + (36*2^(2/3)*(Cosh[3*C[1]] - Sinh[3*C[
1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^18 + 40*x^12*
Cosh[6*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (C
osh[18*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^
6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + (9*2^(1/3)*(32*x^18
 + 40*x^12*Cosh[6*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12
*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + 
(-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^6])])/(
18*x^3)}, {y[x] -> (-Cosh[3*C[1]] + Sinh[3*C[1]] + x^3*Sqrt[36 + Cosh[6*C[1]]/x^
6 - Sinh[6*C[1]]/x^6 + (36*2^(2/3)*(Cosh[3*C[1]] - Sinh[3*C[1]])*((-1 + 2*x^6)*C
osh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^18 + 40*x^12*Cosh[6*C[1]] - x^6*C
osh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[18*C[1]] - Sinh[
18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(
Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + (9*2^(1/3)*(32*x^18 + 40*x^12*Cosh[6*C[
1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[18*C[
1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*Sinh[3
*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^6] + x^3*Sqrt[72 + (2*Cosh[
6*C[1]])/x^6 - (2*Sinh[6*C[1]])/x^6 + (36*2^(2/3)*(-Cosh[3*C[1]] + Sinh[3*C[1]])
*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^18 + 40*x^12*Cosh
[6*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] + (Cosh[
18*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16*x^6)*S
inh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) - (9*2^(1/3)*(32*x^18 + 4
0*x^12*Cosh[6*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1
]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 
+ 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^6 + (2*(432
*x^6*Cosh[3*C[1]] - Cosh[9*C[1]] - 432*x^6*Sinh[3*C[1]] + Sinh[9*C[1]]))/(x^9*Sq
rt[36 + Cosh[6*C[1]]/x^6 - Sinh[6*C[1]]/x^6 + (36*2^(2/3)*(Cosh[3*C[1]] - Sinh[3
*C[1]])*((-1 + 2*x^6)*Cosh[3*C[1]] + (1 + 2*x^6)*Sinh[3*C[1]]))/(32*x^18 + 40*x^
12*Cosh[6*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh[12*C[1]] +
 (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]] + (-1 + 16
*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3) + (9*2^(1/3)*(32*x
^18 + 40*x^12*Cosh[6*C[1]] - x^6*Cosh[12*C[1]] - 40*x^12*Sinh[6*C[1]] + x^6*Sinh
[12*C[1]] + (Cosh[18*C[1]] - Sinh[18*C[1]])*Sqrt[x^12*((1 + 16*x^6)*Cosh[3*C[1]]
 + (-1 + 16*x^6)*Sinh[3*C[1]])^3*(Cosh[21*C[1]] + Sinh[21*C[1]])])^(1/3))/x^6])]
)/(18*x^3)}}

Maple raw input

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

Maple raw output

ln(x)+Intat(1/(2*_a-(_a^2+1)^(1/2)),_a = y(x))+_C1 = 0