23.3.148 problem 150

Internal problem ID [5862]
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 : 150
Date solved : Tuesday, September 30, 2025 at 02:04:49 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 59

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

\[ y = -a \cot \left (\frac {x}{2}\right ) \ln \left (\frac {1}{\cos \left (x \right )+1}\right )+4 \arctan \left (\tan \left (\frac {x}{4}\right )\right ) a +2 \ln \left (\csc \left (\frac {x}{2}\right )\right )+2 \ln \left (\sin \left (\frac {x}{2}\right )\right )+\left (-a \ln \left (2\right )-x -c_1 \right ) \cot \left (\frac {x}{2}\right )+c_2 \]

✓ Mathematica. Time used: 0.493 (sec). Leaf size: 77

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

\begin{align*} y(x)&\to \frac {2 i a \left (1+e^{i x}\right ) \log \left (1+e^{i x}\right )+e^{i x} (2 a x-i x+1+2 c_2)-i x-1+2 i c_1}{-1+e^{i x}} \end{align*}

✗ Sympy

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

Timed Out