2.1222   ODE No. 1222

\[ \left (x^2+1\right ) y''(x)+x y'(x)+2 y(x)=0 \]

✓ Mathematica : cpu = 0.0193397 (sec), leaf count = 50

DSolve[2*y[x] + x*Derivative[1][y][x] + (1 + x^2)*Derivative[2][y][x] == 0,y[x],x]
 

\[\left \{\left \{y(x)\to c_1 \cos \left (\sqrt {2} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+1}}\right )\right )+c_2 \sin \left (\sqrt {2} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+1}}\right )\right )\right \}\right \}\]

✓ Maple : cpu = 0.027 (sec), leaf count = 23

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

\[y \left (x \right ) = c_{1} \sin \left (\sqrt {2}\, \operatorname {arcsinh}\left (x \right )\right )+c_{2} \cos \left (\sqrt {2}\, \operatorname {arcsinh}\left (x \right )\right )\]