64.22.10 problem 4(c)

Internal problem ID [13675]
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 : Tuesday, January 28, 2025 at 08:24:54 PM
CAS classification : [_Lienard]

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

✓ Solution by Maple

Time used: 0.431 (sec). Leaf size: 29

dsolve(diff(x(t),t$2)-tan(t)*diff(x(t),t)+x(t)=0,x(t), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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