59.12.7 problem 19.1 (vii)

Internal problem ID [15083]
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 : Thursday, October 02, 2025 at 10:02:48 AM
CAS classification : [[_Emden, _Fowler]]

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

With initial conditions

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

✓ Maple. Time used: 0.112 (sec). Leaf size: 17

ode:=4*t^2*diff(diff(x(t),t),t)+8*t*diff(x(t),t)+5*x(t) = 0; 
ic:=[x(1) = 2, D(x)(1) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 232

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

\begin{align*} 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 )} \end{align*}

✓ Sympy. Time used: 0.125 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(4*t**2*Derivative(x(t), (t, 2)) + 8*t*Derivative(x(t), t) + 5*x(t),0) 
ics = {x(1): 2, Subs(Derivative(x(t), t), t, 1): 0} 
dsolve(ode,func=x(t),ics=ics)
 

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