2.336   ODE No. 336

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

✓ Mathematica : cpu = 0.404635 (sec), leaf count = 74

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

\[\text {Solve}\left [a x y(x)+\frac {1}{2} \sqrt {x^2+1} x+\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+1}}\right )+\frac {1}{2} \left (y(x) \sqrt {y(x)^2+1}+\tanh ^{-1}\left (\frac {y(x)}{\sqrt {y(x)^2+1}}\right )\right )=c_1,y(x)\right ]\]

✓ Maple : cpu = 0.043 (sec), leaf count = 41

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

\[\frac {x \sqrt {x^{2}+1}}{2}+\frac {\operatorname {arcsinh}\left (x \right )}{2}+a x y \left (x \right )+\frac {y \left (x \right ) \sqrt {y \left (x \right )^{2}+1}}{2}+\frac {\operatorname {arcsinh}\left (y \left (x \right )\right )}{2}+c_{1} = 0\]