89.2.14 problem 14

Internal problem ID [24279]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 27
Problem number : 14
Date solved : Thursday, October 02, 2025 at 10:06:43 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x \csc \left (\frac {y}{x}\right )-y+x y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 11

ode:=x*csc(y(x)/x)-y(x)+x*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \arccos \left (\ln \left (x \right )+c_1 \right ) x \]

✓ Mathematica. Time used: 0.268 (sec). Leaf size: 30

ode=(x*Csc[y[x]/x]-y[x])+x*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x \arccos (\log (x)-c_1)\\ y(x)&\to x \arccos (\log (x)-c_1) \end{align*}

✓ Sympy. Time used: 0.526 (sec). Leaf size: 24

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

\[ \left [ y{\left (x \right )} = x \left (- \operatorname {acos}{\left (C_{1} + \log {\left (x \right )} \right )} + 2 \pi \right ), \ y{\left (x \right )} = x \operatorname {acos}{\left (C_{1} + \log {\left (x \right )} \right )}\right ] \]