78.4.7 problem 8

Internal problem ID [18059]
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 : 8
Date solved : Monday, March 31, 2025 at 05:01:40 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 13

ode:=-1/y(x)*sin(x/y(x))+x/y(x)^2*sin(x/y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x}{\pi -c_1} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 19

ode=(-1/y[x]*Sin[x/y[x]])+(x/y[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 c_1 x \\ y(x)\to \text {ComplexInfinity} \\ y(x)\to \text {ComplexInfinity} \\ \end{align*}

✓ Sympy. Time used: 0.202 (sec). Leaf size: 5

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

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