54.3.406 problem 1423

Internal problem ID [12701]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1423
Date solved : Friday, October 03, 2025 at 03:46:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.214 (sec). Leaf size: 110

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

\[ y = \frac {\sqrt {\cos \left (x \right )}\, \left (\frac {\cos \left (2 x \right )}{2}-\frac {1}{2}\right )^{\frac {1}{2}+\frac {\sqrt {1-4 a}}{4}} \left (\cos \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {\sqrt {1-4 a}}{4}+\frac {3}{4}, \frac {\sqrt {1-4 a}}{4}+\frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_2 +\operatorname {hypergeom}\left (\left [\frac {\sqrt {1-4 a}}{4}+\frac {1}{4}, \frac {\sqrt {1-4 a}}{4}+\frac {1}{4}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) c_1 \right )}{\sqrt {\sin \left (2 x \right )}} \]

✓ Mathematica. Time used: 0.059 (sec). Leaf size: 61

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

\begin{align*} 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 ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x)/sin(x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve a*y(x)/sin(x)**2 + Derivative(y(x), (x, 2))