23.3.243 problem 245

Internal problem ID [5957]
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 : 245
Date solved : Tuesday, September 30, 2025 at 02:06:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y-x y^{\prime }+\left (1-x \cot \left (x \right )\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.179 (sec). Leaf size: 15

ode:=y(x)-x*diff(y(x),x)+(1-x*cot(x))*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,\operatorname {csgn}\left (\sec \left (x \right )\right ) \sin \left (x \right )+c_2 x \]

✓ Mathematica. Time used: 0.086 (sec). Leaf size: 15

ode=y[x] - x*D[y[x],x] + (1 - x*Cot[x])*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 x+c_2 \sin (x) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + (-x/tan(x) + 1)*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x*Derivative(y(x), (x, 2))/tan(x) + y(x)