2.1796   ODE No. 1796

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

✓ Mathematica : cpu = 0.334922 (sec), leaf count = 130

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

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

✓ Maple : cpu = 0.297 (sec), leaf count = 51

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

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