2.1423   ODE No. 1423

\[ y''(x)=-a y(x) \csc ^2(x) \]

✓ Mathematica : cpu = 0.0588137 (sec), leaf count = 70

DSolve[Derivative[2][y][x] == -(a*Csc[x]^2*y[x]),y[x],x]
 

\[\left \{\left \{y(x)\to c_1 \sqrt [4]{\cos ^2(x)-1} P_{-\frac {1}{2}}^{\frac {1}{2} \sqrt {1-4 a}}(\cos (x))+c_2 \sqrt [4]{\cos ^2(x)-1} Q_{-\frac {1}{2}}^{\frac {1}{2} \sqrt {1-4 a}}(\cos (x))\right \}\right \}\]

✓ Maple : cpu = 0.441 (sec), leaf count = 110

dsolve(diff(diff(y(x),x),x) = -a/sin(x)^2*y(x),y(x))
 

\[y \left (x \right ) = \frac {\sqrt {\cos \left (x \right )}\, \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {1}{2}+\frac {\sqrt {1-4 a}}{4}} \left (\cos \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {\sqrt {1-4 a}}{4}+\frac {3}{4}, \frac {\sqrt {1-4 a}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{2} +\operatorname {hypergeom}\left (\left [\frac {\sqrt {1-4 a}}{4}+\frac {1}{4}, \frac {\sqrt {1-4 a}}{4}+\frac {1}{4}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_{1} \right )}{\sqrt {\sin \left (2 x \right )}}\]