20.5.2 problem Problem 2

Internal problem ID [3685]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.9, Exact Differential Equations. page 91
Problem number : Problem 2
Date solved : Tuesday, March 04, 2025 at 05:07:40 PM
CAS classification : [[_homogeneous, `class G`], _exact]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 5.369 (sec). Leaf size: 34

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

\begin{align*} y(x)\to -\frac {\arccos \left (-\frac {c_1}{x}\right )}{x} \\ y(x)\to \frac {\arccos \left (-\frac {c_1}{x}\right )}{x} \\ \end{align*}

✓ Sympy. Time used: 0.906 (sec). Leaf size: 32

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

\[ \left [ y{\left (x \right )} = - \frac {\operatorname {atan}{\left (\sqrt {C_{1} x^{2} - 1} \right )}}{x}, \ y{\left (x \right )} = \frac {\operatorname {atan}{\left (\sqrt {C_{1} x^{2} - 1} \right )}}{x}\right ] \]