2.1756   ODE No. 1756

\[ a y(x) y''(x)+b y'(x)^2-\frac {y(x) y'(x)}{\sqrt {c^2+x^2}}=0 \]

✓ Mathematica : cpu = 0.50173 (sec), leaf count = 124

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

\[\left \{\left \{y(x)\to c_2 \exp \left (\int _1^x\frac {e^{\frac {\tanh ^{-1}\left (\frac {K[2]}{\sqrt {c^2+K[2]^2}}\right )}{a}}}{c_1-\int _1^{K[2]}\frac {e^{\frac {\tanh ^{-1}\left (\frac {K[1]}{\sqrt {c^2+K[1]^2}}\right )}{a}} \left (-\sqrt {c^2+K[1]^2} a-b \sqrt {c^2+K[1]^2}\right )}{a \sqrt {c^2+K[1]^2}}dK[1]}dK[2]\right )\right \}\right \}\]

✓ Maple : cpu = 0.273 (sec), leaf count = 78

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

\[y \left (x \right ) = {\left (\frac {a \left (a +1\right )}{\left (a +b \right ) \left (c_{1} 2^{\frac {1}{a}} a \,x^{\frac {a +1}{a}} \operatorname {hypergeom}\left (\left [-\frac {1}{2 a}, -\frac {a +1}{2 a}\right ], \left [\frac {a -1}{a}\right ], -\frac {c^{2}}{x^{2}}\right )+c_{2} a +c_{2} \right )}\right )}^{-\frac {a}{a +b}}\]