23.1.213 problem 209

Internal problem ID [4820]
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 : 209
Date solved : Tuesday, September 30, 2025 at 08:42:53 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.076 (sec). Leaf size: 31

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

\[ \frac {x \sin \left (\frac {2 y}{x}\right )+2 y}{4 x}-\ln \left (x \right )-c_1 = 0 \]

✓ Mathematica. Time used: 0.166 (sec). Leaf size: 31

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

\[ \text {Solve}\left [\frac {y(x)}{2 x}+\frac {1}{4} \sin \left (\frac {2 y(x)}{x}\right )=\log (x)+c_1,y(x)\right ] \]

✗ Sympy

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

TypeError : cannot determine truth value of Relational: -4*x/(exp(2*_X0*I/x) + 2*exp(_X0*I/x) + 1) -