60.3.417 problem 1423

Internal problem ID [11427]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1423
Date solved : Tuesday, January 28, 2025 at 06:05:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {a y}{\sin \left (x \right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.957 (sec). Leaf size: 110

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

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

✓ Solution by Mathematica

Time used: 0.099 (sec). Leaf size: 61

DSolve[D[y[x],{x,2}] == -(a*Csc[x]^2*y[x]),y[x],x,IncludeSingularSolutions -> True]
 

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