30.2.22 problem 22

Internal problem ID [7450]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.3, Linear equations. Exercises. page 54
Problem number : 22
Date solved : Tuesday, September 30, 2025 at 04:35:45 PM
CAS classification : [_linear]

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

With initial conditions

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 13

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

\[ y = 1-x \cot \left (x \right )+\csc \left (x \right ) \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 27

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

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

✓ Sympy. Time used: 0.611 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = - \frac {x}{\tan {\left (x \right )}} + 1 + \frac {1}{\sin {\left (x \right )}} \]