Internal problem ID [11746]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {4 t^{2} x^{\prime \prime }+8 x^{\prime } t +5 x=0} \] With initial conditions \begin {align*} [x \left (1\right ) = 2, x^{\prime }\left (1\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.015 (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: 0.101 (sec). Leaf size: 232
DSolve[{4*t^2*x''[t]+8*t*x[t]+5*x[t]==0,{x[1]==2,x'[1]==0}},x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {\sqrt {t} \left (\left (2 \operatorname {BesselJ}\left (-1+2 i,2 \sqrt {2}\right )+\sqrt {2} \operatorname {BesselJ}\left (2 i,2 \sqrt {2}\right )-2 \operatorname {BesselJ}\left (1+2 i,2 \sqrt {2}\right )\right ) \operatorname {BesselJ}\left (-2 i,2 \sqrt {2} \sqrt {t}\right )-\left (2 \operatorname {BesselJ}\left (-1-2 i,2 \sqrt {2}\right )+\sqrt {2} \operatorname {BesselJ}\left (-2 i,2 \sqrt {2}\right )-2 \operatorname {BesselJ}\left (1-2 i,2 \sqrt {2}\right )\right ) \operatorname {BesselJ}\left (2 i,2 \sqrt {2} \sqrt {t}\right )\right )}{\operatorname {BesselJ}\left (-1+2 i,2 \sqrt {2}\right ) \operatorname {BesselJ}\left (-2 i,2 \sqrt {2}\right )-\operatorname {BesselJ}\left (-1-2 i,2 \sqrt {2}\right ) \operatorname {BesselJ}\left (2 i,2 \sqrt {2}\right )+\operatorname {BesselJ}\left (2 i,2 \sqrt {2}\right ) \operatorname {BesselJ}\left (1-2 i,2 \sqrt {2}\right )-\operatorname {BesselJ}\left (-2 i,2 \sqrt {2}\right ) \operatorname {BesselJ}\left (1+2 i,2 \sqrt {2}\right )} \]