65.4.22 problem 22

Internal problem ID [15681]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.2, page 53
Problem number : 22
Date solved : Thursday, October 02, 2025 at 10:23:01 AM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (\frac {\pi }{2}\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 11

ode:=diff(y(x),x) = cot(x)*y(x)+csc(x); 
ic:=[y(1/2*Pi) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\cos \left (x \right )+\sin \left (x \right ) \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to \sin (x) \left (\int _{\frac {\pi }{2}}^x\csc ^2(K[1])dK[1]+1\right ) \end{align*}

✓ Sympy. Time used: 0.560 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = \sin {\left (x \right )} - \cos {\left (x \right )} \]