29.8.3 problem 208

Internal problem ID [4808]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 8
Problem number : 208
Date solved : Sunday, March 30, 2025 at 03:59:13 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 44

ode:=x*diff(y(x),x) = y(x)+x*sin(y(x)/x); 
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: 0.323 (sec). Leaf size: 52

ode=x D[y[x],x]==y[x]+x Sin[y[x]/x]; 
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: 1.224 (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)
 

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