ODE No. 1423

$$y^{″}(x)=-ay(x){\csc }^2(x)$$ ✓ Mathematica : cpu = 0.0562062 (sec), leaf count = 70

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

$$\left\{\{y(x)\to c_1\sqrt[4]{{\cos }^2(x)-1}{P}_{-\frac{1}{2}}^{\frac{1}{2}\sqrt{1-4a}}(\cos (x))+c_2\sqrt[4]{{\cos }^2(x)-1}{Q}_{-\frac{1}{2}}^{\frac{1}{2}\sqrt{1-4a}}(\cos (x))\}\right\}$$ ✓ Maple : cpu = 0.25 (sec), leaf count = 132

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

$$y\left(x\right)=\frac{\sqrt{-2\cos \left(2x\right)+2}{(2\cos \left(2x\right)+2)}^{\frac{1}{4}}{(\frac{\cos \left(2x\right)}{2}-\frac{1}{2})}^{\frac{\sqrt{1-4a}}{4}}(\mathrm{hypergeom}([\frac{\sqrt{1-4a}}{4}+\frac{3}{4},\frac{\sqrt{1-4a}}{4}+\frac{3}{4}],\left[\frac{3}{2}\right],\frac{\cos \left(2x\right)}{2}+\frac{1}{2})\sqrt{2\cos \left(2x\right)+2}{c}_2+\mathrm{hypergeom}([\frac{\sqrt{1-4a}}{4}+\frac{1}{4},\frac{\sqrt{1-4a}}{4}+\frac{1}{4}],\left[\frac{1}{2}\right],\frac{\cos \left(2x\right)}{2}+\frac{1}{2})c_1)}{\sqrt{\sin \left(2x\right)}}$$