62.12.10 problem Ex 11

Internal problem ID [12785]
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 19. Summary. Page 29
Problem number : Ex 11
Date solved : Monday, March 31, 2025 at 07:07:20 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 14

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

\[ y = x \arcsin \left (\frac {1}{x c_1}\right ) \]

✓ Mathematica. Time used: 12.405 (sec). Leaf size: 21

ode=(x*Sin[y[x]/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 \left (\frac {e^{c_1}}{x}\right ) \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.253 (sec). Leaf size: 19

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

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