44.5.19 problem 7(a)

Internal problem ID [9175]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.7. Homogeneous Equations. Page 28
Problem number : 7(a)
Date solved : Tuesday, September 30, 2025 at 06:11:58 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 25

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

\[ y = \operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {1}{-\sin \left (\textit {\_a} \right )+\cos \left (\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} +\ln \left (x \right )+c_1 \right ) x \]

✓ Mathematica. Time used: 0.215 (sec). Leaf size: 36

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {1}{\cos (K[1])+K[1]-\sin (K[1])}dK[1]=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 53.460 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = C_{1} e^{- \sqrt {2} \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\cos {\left (\frac {\pi }{4} + \frac {1}{u_{1}} \right )}}{\sqrt {2} u_{1} \cos {\left (\frac {\pi }{4} + \frac {1}{u_{1}} \right )} + 1}\, du_{1}} \]