23.1.210 problem 206

Internal problem ID [4817]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 206
Date solved : Tuesday, September 30, 2025 at 08:42:35 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x y^{\prime }&=y-x \cot \left (\frac {y}{x}\right )^{2} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 23

ode:=x*diff(y(x),x) = y(x)-x*cot(y(x)/x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (2 \tan \left (\textit {\_Z} \right )-2 \textit {\_Z} +2 \ln \left (x \right )+\pi +2 c_1 \right ) x \]

✓ Mathematica. Time used: 0.161 (sec). Leaf size: 35

ode=x*D[y[x],x]==y[x]-x*Cot[y[x]/x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {1}{2} \left (\tan \left (\frac {y(x)}{x}\right )-\arctan \left (\tan \left (\frac {y(x)}{x}\right )\right )\right )=-\frac {\log (x)}{2}+c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*cot(y(x)/x)**2 + x*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : cannot determine truth value of Relational: -1 < x