23.3.145 problem 147

Internal problem ID [5859]
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 : 147
Date solved : Friday, October 03, 2025 at 01:44:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.198 (sec). Leaf size: 91

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

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

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-c*y(x) - Derivative(y(x), (x, 2)))*tan(b