56.3.6 problem Ex 6

Internal problem ID [14093]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 10. Homogeneous equations. Page 15
Problem number : Ex 6
Date solved : Thursday, October 02, 2025 at 09:13:03 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

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

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

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

✓ Mathematica. Time used: 0.225 (sec). Leaf size: 13

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

\begin{align*} y(x)&\to x \arcsin (\log (x)+c_1) \end{align*}

✓ Sympy. Time used: 0.558 (sec). Leaf size: 24

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

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