23.3.45 problem 47

Internal problem ID [5759]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 47
Date solved : Friday, October 03, 2025 at 01:43:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} a \csc \left (x \right )^{2} y+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.055 (sec). Leaf size: 40

ode:=a*csc(x)^2*y(x)+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 61

ode=a*Csc[x]^2*y[x] + D[y[x],{x,2}] == 0; 
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))