65.12.7 problem 19.1 (vii)

Internal problem ID [13802]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 19, CauchyEuler equations. Exercises page 174
Problem number : 19.1 (vii)
Date solved : Tuesday, January 28, 2025 at 06:04:01 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 4 t^{2} x^{\prime \prime }+8 t x^{\prime }+5 x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (1\right )&=2\\ x^{\prime }\left (1\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 17

dsolve([4*t^2*diff(x(t),t$2)+8*t*diff(x(t),t)+5*x(t)=0,x(1) = 2, D(x)(1) = 0],x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\sin \left (\ln \left (t \right )\right )+2 \cos \left (\ln \left (t \right )\right )}{\sqrt {t}} \]

✓ Solution by Mathematica

Time used: 0.066 (sec). Leaf size: 232

DSolve[{4*t^2*D[x[t],{t,2}]+8*t*x[t]+5*x[t]==0,{x[1]==2,Derivative[1][x][1 ]==0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {\sqrt {t} \left (\left (2 \operatorname {BesselJ}\left (-1+2 i,2 \sqrt {2}\right )+\sqrt {2} \operatorname {BesselJ}\left (2 i,2 \sqrt {2}\right )-2 \operatorname {BesselJ}\left (1+2 i,2 \sqrt {2}\right )\right ) \operatorname {BesselJ}\left (-2 i,2 \sqrt {2} \sqrt {t}\right )-\left (2 \operatorname {BesselJ}\left (-1-2 i,2 \sqrt {2}\right )+\sqrt {2} \operatorname {BesselJ}\left (-2 i,2 \sqrt {2}\right )-2 \operatorname {BesselJ}\left (1-2 i,2 \sqrt {2}\right )\right ) \operatorname {BesselJ}\left (2 i,2 \sqrt {2} \sqrt {t}\right )\right )}{\operatorname {BesselJ}\left (-1+2 i,2 \sqrt {2}\right ) \operatorname {BesselJ}\left (-2 i,2 \sqrt {2}\right )-\operatorname {BesselJ}\left (-1-2 i,2 \sqrt {2}\right ) \operatorname {BesselJ}\left (2 i,2 \sqrt {2}\right )+\operatorname {BesselJ}\left (2 i,2 \sqrt {2}\right ) \operatorname {BesselJ}\left (1-2 i,2 \sqrt {2}\right )-\operatorname {BesselJ}\left (-2 i,2 \sqrt {2}\right ) \operatorname {BesselJ}\left (1+2 i,2 \sqrt {2}\right )} \]