65.12.3 problem 19.1 (iii)

Internal problem ID [13798]
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)
Date solved : Tuesday, January 28, 2025 at 06:03:48 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t^{2} x^{\prime \prime }-5 t x^{\prime }+10 x&=0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.033 (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: 0.117 (sec). Leaf size: 256

DSolve[{t^2*D[x[t],{t,2}]-5*t*x[t]+10*x[t]==0,{x[1]==2,Derivative[1][x][1 ]==1}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {2 \sqrt {t} \left (\left (\operatorname {BesselI}\left (-1-i \sqrt {39},2 \sqrt {5}\right )+\operatorname {BesselI}\left (1-i \sqrt {39},2 \sqrt {5}\right )\right ) \operatorname {BesselI}\left (i \sqrt {39},2 \sqrt {5} \sqrt {t}\right )-\left (\operatorname {BesselI}\left (-1+i \sqrt {39},2 \sqrt {5}\right )+\operatorname {BesselI}\left (1+i \sqrt {39},2 \sqrt {5}\right )\right ) \operatorname {BesselI}\left (-i \sqrt {39},2 \sqrt {5} \sqrt {t}\right )\right )}{\operatorname {BesselI}\left (i \sqrt {39},2 \sqrt {5}\right ) \left (\operatorname {BesselI}\left (-1-i \sqrt {39},2 \sqrt {5}\right )+\operatorname {BesselI}\left (1-i \sqrt {39},2 \sqrt {5}\right )\right )-\operatorname {BesselI}\left (-i \sqrt {39},2 \sqrt {5}\right ) \left (\operatorname {BesselI}\left (-1+i \sqrt {39},2 \sqrt {5}\right )+\operatorname {BesselI}\left (1+i \sqrt {39},2 \sqrt {5}\right )\right )} \]