78.1.13 problem 2.c

Internal problem ID [20939]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 1, First order ODEs. Problems section 1.5
Problem number : 2.c
Date solved : Thursday, October 02, 2025 at 06:49:41 PM
CAS classification : [[_homogeneous, `class G`], _exact]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 14

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

\[ y = \frac {\arcsin \left (\frac {c_1}{x}\right )}{x} \]

✓ Mathematica. Time used: 2.568 (sec). Leaf size: 16

ode=(Sin[x*y[x]]+x*y[x]*Cos[x*y[x]])+(x^2*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 (\frac {c_1}{x}\right )}{x} \end{align*}

✓ Sympy. Time used: 0.955 (sec). Leaf size: 36

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

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