15.2.23 problem 23

Internal problem ID [2893]
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 : 23
Date solved : Sunday, March 30, 2025 at 12:46:39 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.276 (sec). Leaf size: 103

ode:=diff(y(x),x) = y(x)/x+tanh(y(x)/x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \operatorname {arctanh}\left (\frac {-c_1 \,x^{2}+\sqrt {c_1 \,x^{2} \left (c_1 \,x^{2}-1\right )}}{-c_1 \,x^{2}+\sqrt {c_1 \,x^{2} \left (c_1 \,x^{2}-1\right )}+1}\right ) x \\ y &= \operatorname {arctanh}\left (\frac {c_1 \,x^{2}+\sqrt {c_1 \,x^{2} \left (c_1 \,x^{2}-1\right )}}{c_1 \,x^{2}-1+\sqrt {c_1 \,x^{2} \left (c_1 \,x^{2}-1\right )}}\right ) x \\ \end{align*}

✓ Mathematica. Time used: 1.974 (sec). Leaf size: 19

ode=D[y[x],x]==y[x]/x+Tanh[y[x]/x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to x \text {arcsinh}\left (e^{c_1} x\right ) \\ y(x)\to 0 \\ \end{align*}

✗ Sympy

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

RecursionError : maximum recursion depth exceeded