36.2.22 problem 22

Internal problem ID [6315]
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 : Wednesday, March 05, 2025 at 12:33:56 AM
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: 0.011 (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,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 14

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]
 

\[ y(x)\to -x \cot (x)+\csc (x)+1 \]

✓ Sympy. Time used: 0.939 (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)
 

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