19.3.3 problem 11

Internal problem ID [3546]
Book : Differential equations and linear algebra, Stephen W. Goode, second edition, 2000
Section : 1.8, page 68
Problem number : 11
Date solved : Tuesday, March 04, 2025 at 04:45:17 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 26.462 (sec). Leaf size: 56

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

\begin{align*} y(x)\to -x \arccos \left (\frac {e^{-c_1}}{x}\right ) \\ y(x)\to x \arccos \left (\frac {e^{-c_1}}{x}\right ) \\ y(x)\to -\frac {\pi x}{2} \\ y(x)\to \frac {\pi x}{2} \\ \end{align*}

✓ Sympy. Time used: 1.283 (sec). Leaf size: 32

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

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