65.7.7 problem 7

Internal problem ID [15715]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.3.3, page 71
Problem number : 7
Date solved : Thursday, October 02, 2025 at 10:23:53 AM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (\frac {\pi }{2}\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 12

ode:=diff(y(x),x) = cot(x)*y(x)+sin(x); 
ic:=[y(1/2*Pi) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \left (x -\frac {\pi }{2}\right ) \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 16

ode=D[y[x],x]==Cot[x]*y[x]+Sin[x]; 
ic={y[Pi/2]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{2} (\pi -2 x) \sin (x) \end{align*}

✓ Sympy. Time used: 0.389 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = \left (x - \frac {\pi }{2}\right ) \sin {\left (x \right )} \]