64.22.9 problem 4(b)

Internal problem ID [13674]
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(b)
Date solved : Tuesday, January 28, 2025 at 08:24:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (t^{4}+t^{2}\right ) x^{\prime \prime }+2 t^{3} x^{\prime }+3 x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.767 (sec). Leaf size: 85

dsolve((t^4+t^2)*diff(x(t),t$2)+2*t^3*diff(x(t),t)+3*x(t)=0,x(t), singsol=all)
 

\[ x \left (t \right ) = \sqrt {t}\, \left (c_{1} t^{-\frac {i \sqrt {11}}{2}} \operatorname {hypergeom}\left (\left [\frac {3}{4}-\frac {i \sqrt {11}}{4}, \frac {1}{4}-\frac {i \sqrt {11}}{4}\right ], \left [1-\frac {i \sqrt {11}}{2}\right ], -t^{2}\right )+c_{2} t^{\frac {i \sqrt {11}}{2}} \operatorname {hypergeom}\left (\left [\frac {3}{4}+\frac {i \sqrt {11}}{4}, \frac {1}{4}+\frac {i \sqrt {11}}{4}\right ], \left [1+\frac {i \sqrt {11}}{2}\right ], -t^{2}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.158 (sec). Leaf size: 139

DSolve[(t^4+t^2)*D[x[t],{t,2}]+2*t^3*D[x[t],t]+3*x[t]==0,{x[t]},t,IncludeSingularSolutions -> True]
 

\[ x(t)\to t^{\frac {1}{2}-\frac {i \sqrt {11}}{2}} \left (c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4}-\frac {i \sqrt {11}}{4},\frac {3}{4}-\frac {i \sqrt {11}}{4},1-\frac {i \sqrt {11}}{2},-t^2\right )+c_2 t^{i \sqrt {11}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4}+\frac {i \sqrt {11}}{4},\frac {3}{4}+\frac {i \sqrt {11}}{4},1+\frac {i \sqrt {11}}{2},-t^2\right )\right ) \]