26.1.13 problem First order with homogeneous Coefficients. Exercise 7.14, page 61

Internal problem ID [6918]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 7
Problem number : First order with homogeneous Coefficients. Exercise 7.14, page 61
Date solved : Tuesday, September 30, 2025 at 04:05:37 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.091 (sec). Leaf size: 22

ode:=diff(y(x),x)-y(x)/x+csc(y(x)/x) = 0; 
ic:=[y(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= \arccos \left (\ln \left (x \right )+1\right ) x \\ y &= -\arccos \left (\ln \left (x \right )+1\right ) x \\ \end{align*}

✓ Mathematica. Time used: 0.257 (sec). Leaf size: 24

ode=D[y[x],x]-y[x]/x+Csc[y[x]/x]==0; 
ic=y[1]==0; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x \arccos (\log (x)+1)\\ y(x)&\to x \arccos (\log (x)+1) \end{align*}

✓ Sympy. Time used: 0.561 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = x \operatorname {acos}{\left (\log {\left (x \right )} + 1 \right )} \]