85.13.10 problem 10

Internal problem ID [22516]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. B Exercises at page 40
Problem number : 10
Date solved : Thursday, October 02, 2025 at 08:44:52 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.476 (sec). Leaf size: 121

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

\[ y = \arctan \left (\frac {\operatorname {RootOf}\left (x^{4} \textit {\_Z}^{4}+2 x^{4} \textit {\_Z}^{3}-2 x^{4} \textit {\_Z} +\textit {\_Z}^{2} c_1^{2}-x^{4}\right ) c_1}{x^{2} \left (\operatorname {RootOf}\left (x^{4} \textit {\_Z}^{4}+2 x^{4} \textit {\_Z}^{3}-2 x^{4} \textit {\_Z} +\textit {\_Z}^{2} c_1^{2}-x^{4}\right )+1\right )}, \operatorname {RootOf}\left (x^{4} \textit {\_Z}^{4}+2 x^{4} \textit {\_Z}^{3}-2 x^{4} \textit {\_Z} +\textit {\_Z}^{2} c_1^{2}-x^{4}\right )\right ) x \]

✓ Mathematica. Time used: 0.997 (sec). Leaf size: 78

ode=(2*x*Sin[y[x]/x]+2*x*Tan[y[x]/x]-y[x]*Cos[y[x]/x ] - y[x]*Sec[y[x]/x]^2  ) +( x*Cos[y[x]/x]+x*Sec[y[x]/x]^2 )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\pi x\\ y(x)&\to \pi x\\ \text {Solve}\left [\frac {3}{2} \text {arctanh}\left (\cos \left (\frac {2 y(x)}{x}\right )\right )+\log \left (\sin \left (\frac {y(x)}{2 x}\right )\right )-\frac {1}{2} \log \left (\sin \left (\frac {2 y(x)}{x}\right )\right )-\log \left (\cos \left (\frac {y(x)}{2 x}\right )\right )=2 \log (x)+c_1,y(x)\right ] \end{align*}

✗ Sympy

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

Timed Out