89.3.16 problem 16

Internal problem ID [24314]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 34
Problem number : 16
Date solved : Thursday, October 02, 2025 at 10:17:33 PM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} 2 x +y \cos \left (y x \right )+x \cos \left (y x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 15

ode:=2*x+y(x)*cos(x*y(x))+x*cos(x*y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\arcsin \left (x^{2}+c_1 \right )}{x} \]

✓ Mathematica. Time used: 0.279 (sec). Leaf size: 19

ode=( 2*x+y[x]*Cos[x*y[x]]   )+ (x*Cos[x*y[x]] )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\arcsin \left (x^2-c_1\right )}{x} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) + 2/cos(x*y(x)) + y(x)/x cannot be solved by