15.2.13 problem 13

Internal problem ID [2883]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 6, page 25
Problem number : 13
Date solved : Tuesday, March 04, 2025 at 02:58:41 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.016 (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: 0.322 (sec). Leaf size: 52

ode=D[y[x],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: 1.230 (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)
 

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