64.22.10 problem 4(c)

Internal problem ID [13604]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 11, The nth order homogeneous linear differential equation. Section 11.8, Exercises page 583
Problem number : 4(c)
Date solved : Monday, March 31, 2025 at 08:02:26 AM
CAS classification : [_Lienard]

\begin{align*} x^{\prime \prime }-\tan \left (t \right ) x^{\prime }+x&=0 \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 29

ode:=diff(diff(x(t),t),t)-tan(t)*diff(x(t),t)+x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = c_1 \operatorname {LegendreP}\left (\frac {\sqrt {5}}{2}-\frac {1}{2}, \sin \left (t \right )\right )+c_2 \operatorname {LegendreQ}\left (\frac {\sqrt {5}}{2}-\frac {1}{2}, \sin \left (t \right )\right ) \]

✓ Mathematica. Time used: 0.734 (sec). Leaf size: 79

ode=D[x[t],{t,2}]-Tan[t]*D[x[t],t]+x[t]==0; 
ic={}; 
DSolve[{ode,ic},{x[t]},t,IncludeSingularSolutions->True]
 

\[ x(t)\to c_2 G_{2,2}^{2,0}\left (\cos ^2(t)| \begin {array}{c} \frac {1}{4} \left (3-\sqrt {5}\right ),\frac {1}{4} \left (3+\sqrt {5}\right ) \\ 0,0 \\ \end {array} \right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (1-\sqrt {5}\right ),\frac {1}{4} \left (1+\sqrt {5}\right ),1,\cos ^2(t)\right ) \]

✓ Sympy. Time used: 2.046 (sec). Leaf size: 83

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

\[ x{\left (t \right )} = C_{2} \left (- \frac {t^{4} \tan ^{2}{\left (t \right )}}{24} + \frac {t^{4}}{24} - \frac {t^{3} \tan {\left (t \right )}}{6} - \frac {t^{2}}{2} + 1\right ) + C_{1} t \left (\frac {t^{3} \tan ^{3}{\left (t \right )}}{24} - \frac {t^{3} \tan {\left (t \right )}}{12} + \frac {t^{2} \tan ^{2}{\left (t \right )}}{6} - \frac {t^{2}}{6} + \frac {t \tan {\left (t \right )}}{2} + 1\right ) + O\left (t^{6}\right ) \]