Internal problem ID [11742]
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 (iii).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
\[ \boxed {t^{2} x^{\prime \prime }-5 x^{\prime } t +10 x=0} \] With initial conditions \begin {align*} [x \left (1\right ) = 2, x^{\prime }\left (1\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 19
dsolve([t^2*diff(x(t),t$2)-5*t*diff(x(t),t)+10*x(t)=0,x(1) = 2, D(x)(1) = 1],x(t), singsol=all)
\[ x \left (t \right ) = t^{3} \left (-5 \sin \left (\ln \left (t \right )\right )+2 \cos \left (\ln \left (t \right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.197 (sec). Leaf size: 256
DSolve[{t^2*x''[t]-5*t*x[t]+10*x[t]==0,{x[1]==2,x'[1]==1}},x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \frac {2 \sqrt {t} \left (\left (\operatorname {BesselI}\left (-1-i \sqrt {39},2 \sqrt {5}\right )+\operatorname {BesselI}\left (1-i \sqrt {39},2 \sqrt {5}\right )\right ) \operatorname {BesselI}\left (i \sqrt {39},2 \sqrt {5} \sqrt {t}\right )-\left (\operatorname {BesselI}\left (-1+i \sqrt {39},2 \sqrt {5}\right )+\operatorname {BesselI}\left (1+i \sqrt {39},2 \sqrt {5}\right )\right ) \operatorname {BesselI}\left (-i \sqrt {39},2 \sqrt {5} \sqrt {t}\right )\right )}{\operatorname {BesselI}\left (i \sqrt {39},2 \sqrt {5}\right ) \left (\operatorname {BesselI}\left (-1-i \sqrt {39},2 \sqrt {5}\right )+\operatorname {BesselI}\left (1-i \sqrt {39},2 \sqrt {5}\right )\right )-\operatorname {BesselI}\left (-i \sqrt {39},2 \sqrt {5}\right ) \left (\operatorname {BesselI}\left (-1+i \sqrt {39},2 \sqrt {5}\right )+\operatorname {BesselI}\left (1+i \sqrt {39},2 \sqrt {5}\right )\right )} \]