32.1.7 problem First order with homogeneous Coefficients. Exercise 7.8, page 61

Internal problem ID [5777]
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 7.8, page 61
Date solved : Tuesday, March 04, 2025 at 11:42:59 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.059 (sec). Leaf size: 15

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

\[ y \left (x \right ) = \operatorname {RootOf}\left (\textit {\_Z} x c_{1} \sin \left (\textit {\_Z} \right )-1\right ) x \]

✓ Mathematica. Time used: 0.266 (sec). Leaf size: 27

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

\[ \text {Solve}\left [\log \left (\frac {y(x)}{x}\right )+\log \left (\sin \left (\frac {y(x)}{x}\right )\right )=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 9.013 (sec). Leaf size: 17

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

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