54.3.418 problem 1435

Internal problem ID [12713]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1435
Date solved : Friday, October 03, 2025 at 03:47:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {4 \sin \left (3 x \right ) y}{\sin \left (x \right )^{3}} \end{align*}

✓ Maple. Time used: 0.080 (sec). Leaf size: 32

ode:=diff(diff(y(x),x),x) = -4*sin(3*x)/sin(x)^3*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {\sin \left (x \right )}\, \left (c_1 \operatorname {LegendreP}\left (-\frac {1}{2}+4 i, \frac {i \sqrt {47}}{2}, \cos \left (x \right )\right )+c_2 \operatorname {LegendreQ}\left (-\frac {1}{2}+4 i, \frac {i \sqrt {47}}{2}, \cos \left (x \right )\right )\right ) \]

✓ Mathematica. Time used: 0.117 (sec). Leaf size: 61

ode=D[y[x],{x,2}] == -4*Csc[x]^3*Sin[3*x]*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}+4 i}^{\frac {i \sqrt {47}}{2}}(\cos (x))+c_2 Q_{-\frac {1}{2}+4 i}^{\frac {i \sqrt {47}}{2}}(\cos (x))\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x)*sin(3*x)/sin(x)**3 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve 4*y(x)*sin(3*x)/sin(x)**3 + Derivative(y(x), (x, 2))