26.1.4 problem First order with homogeneous Coefficients. Exercise 7.5, page 61

Internal problem ID [6909]
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 7.5, page 61
Date solved : Tuesday, September 30, 2025 at 04:04:43 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 44

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

\[ y = \arctan \left (\frac {2 x c_1}{c_1^{2} x^{2}+1}, \frac {-c_1^{2} x^{2}+1}{c_1^{2} x^{2}+1}\right ) x \]

✓ Mathematica. Time used: 0.207 (sec). Leaf size: 52

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

\begin{align*} y(x)&\to -x \arccos (-\tanh (\log (x)+c_1))\\ y(x)&\to x \arccos (-\tanh (\log (x)+c_1))\\ y(x)&\to 0\\ y(x)&\to -\pi x\\ y(x)&\to \pi x \end{align*}

✓ Sympy. Time used: 0.784 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = 2 x \operatorname {atan}{\left (C_{1} x \right )} \]