65.12.3 problem 19.1 (iii)

Internal problem ID [13719]
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 : Wednesday, March 05, 2025 at 10:13:44 PM
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*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 19

ode:=t^2*diff(diff(x(t),t),t)-5*t*diff(x(t),t)+10*x(t) = 0; 
ic:=x(1) = 2, D(x)(1) = 1; 
dsolve([ode,ic],x(t), singsol=all);
 

\[ x \left (t \right ) = t^{3} \left (-5 \sin \left (\ln \left (t \right )\right )+2 \cos \left (\ln \left (t \right )\right )\right ) \]

✓ Mathematica. Time used: 0.117 (sec). Leaf size: 256

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

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

✓ Sympy. Time used: 0.185 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t**2*Derivative(x(t), (t, 2)) - 5*t*Derivative(x(t), t) + 10*x(t),0) 
ics = {x(1): 2, Subs(Derivative(x(t), t), t, 1): 1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = t^{3} \left (- 5 \sin {\left (\log {\left (t \right )} \right )} + 2 \cos {\left (\log {\left (t \right )} \right )}\right ) \]