73.5.7 problem 6.3 (c)

Internal problem ID [15039]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 6. Simplifying through simplifiction. Additional exercises. page 114
Problem number : 6.3 (c)
Date solved : Thursday, March 13, 2025 at 05:31:03 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} \cos \left (\frac {y}{x}\right ) \left (y^{\prime }-\frac {y}{x}\right )&=1+\sin \left (\frac {y}{x}\right ) \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 12

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

\[ y = \arcsin \left (c_{1} x -1\right ) x \]

✓ Mathematica. Time used: 60.355 (sec). Leaf size: 185

ode=Cos[y[x]/x]*(D[y[x],x]-y[x]/x)==1+Sin[y[x]/x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {\pi x}{2} \\ y(x)\to \frac {3 \pi x}{2} \\ y(x)\to -2 x \arccos \left (\frac {1}{2} \left (e^{\frac {c_1}{2}} \sqrt {x}-\sqrt {2-e^{c_1} x}\right )\right ) \\ y(x)\to 2 x \arccos \left (\frac {1}{2} \left (e^{\frac {c_1}{2}} \sqrt {x}-\sqrt {2-e^{c_1} x}\right )\right ) \\ y(x)\to -2 x \arccos \left (\frac {1}{2} \left (e^{\frac {c_1}{2}} \sqrt {x}+\sqrt {2-e^{c_1} x}\right )\right ) \\ y(x)\to 2 x \arccos \left (\frac {1}{2} \left (e^{\frac {c_1}{2}} \sqrt {x}+\sqrt {2-e^{c_1} x}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 2.000 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((Derivative(y(x), x) - y(x)/x)*cos(y(x)/x) - sin(y(x)/x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = x \left (\pi - \operatorname {asin}{\left (C_{1} x - 1 \right )}\right ), \ y{\left (x \right )} = x \operatorname {asin}{\left (C_{1} x - 1 \right )}\right ] \]