23.3.142 problem 144

Internal problem ID [5856]
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 : 144
Date solved : Friday, October 03, 2025 at 01:44:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.279 (sec). Leaf size: 399

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

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

✓ Mathematica. Time used: 136.863 (sec). Leaf size: 1066280

ode=(a*Cot[x]^2 + b*Cot[x]*Csc[x] + c*Csc[x]^2)*y[x] + k*Cot[x]*D[y[x],x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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✗ Sympy

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

Timed Out