78.4.4 problem 5

Internal problem ID [18056]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 8 (Exact Equations). Problems at page 72
Problem number : 5
Date solved : Monday, March 31, 2025 at 05:01:02 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 17

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

\begin{align*} y &= \frac {\pi }{x} \\ y &= \frac {c_1}{x} \\ \end{align*}

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 49

ode=(y[x]+y[x]*Cos[x*y[x]])+(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 {\pi }{x} \\ y(x)\to \frac {\pi }{x} \\ y(x)\to \frac {c_1}{x} \\ y(x)\to -\frac {\pi }{x} \\ y(x)\to \frac {\pi }{x} \\ \end{align*}

✓ Sympy. Time used: 0.260 (sec). Leaf size: 5

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

\[ y{\left (x \right )} = \frac {C_{1}}{x} \]