20.4.34 problem Problem 50

Internal problem ID [3669]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 50
Date solved : Tuesday, March 04, 2025 at 05:05:20 PM
CAS classification : [_separable]

\begin{align*} \left (1-\sqrt {3}\right ) y^{\prime }+y \sec \left (x \right )&=y^{\sqrt {3}} \sec \left (x \right ) \end{align*}

✓ Maple. Time used: 0.252 (sec). Leaf size: 23

ode:=(1-3^(1/2))*diff(y(x),x)+y(x)*sec(x) = y(x)^(3^(1/2))*sec(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = \left (-\tan \left (x \right ) c_{1} +1+\sec \left (x \right ) c_{1} \right )^{-\frac {1}{2}-\frac {\sqrt {3}}{2}} \]

✓ Mathematica. Time used: 0.57 (sec). Leaf size: 76

ode=(1-Sqrt[3])*D[y[x],x]+y[x]*Sec[x]==y[x]^Sqrt[3]*Sec[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {\log \left (1-\text {$\#$1}^{\sqrt {3}-1}\right )-\left (\sqrt {3}-1\right ) \log (\text {$\#$1})}{\sqrt {3}-1}\&\right ]\left [-\frac {2 \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )}{\sqrt {3}-1}+c_1\right ] \\ y(x)\to 0 \\ y(x)\to 1 \\ \end{align*}

✓ Sympy. Time used: 59.918 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)/cos(x) - y(x)**(sqrt(3))/cos(x) + (1 - sqrt(3))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \frac {- \log {\left (\sin {\left (x \right )} - 1 \right )} + \log {\left (\sin {\left (x \right )} + 1 \right )}}{2 \left (-1 + \sqrt {3}\right )} + \frac {\sqrt {3} \log {\left (- y{\left (x \right )} + y^{\sqrt {3}}{\left (x \right )} \right )}}{-3 + \sqrt {3}} - \frac {3 \log {\left (y{\left (x \right )} \right )}}{-3 + \sqrt {3}} = C_{1} \]