41.2.9 problem 9

Internal problem ID [8711]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.2 Homogeneous equations problems. page 12
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 05:43:09 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.059 (sec). Leaf size: 27

ode:=x*diff(y(x),x) = y(x)*cos(y(x)/x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (\ln \left (x \right )+c_1 -\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \left (-1+\cos \left (\textit {\_a} \right )\right )}d \textit {\_a} \right ) x \]

✓ Mathematica. Time used: 1.195 (sec). Leaf size: 33

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{(\cos (K[1])-1) K[1]}dK[1]=\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 2.168 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\cos {\left (\frac {1}{u_{1}} \right )}}{u_{1} \left (\cos {\left (\frac {1}{u_{1}} \right )} - 1\right )}\, du_{1}} \]