65.12.7 problem 19.1 (vii)
Internal
problem
ID
[13723]
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
:
Wednesday, March 05, 2025 at 10:13:57 PM
CAS
classification
:
[[_Emden, _Fowler]]
\begin{align*} 4 t^{2} x^{\prime \prime }+8 t x^{\prime }+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: 0.027 (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,ic],x(t), singsol=all);
\[
x \left (t \right ) = \frac {\sin \left (\ln \left (t \right )\right )+2 \cos \left (\ln \left (t \right )\right )}{\sqrt {t}}
\]
✓ Mathematica. Time used: 0.066 (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]
\[
x(t)\to \frac {\sqrt {t} \left (\left (2 \operatorname {BesselJ}\left (-1+2 i,2 \sqrt {2}\right )+\sqrt {2} \operatorname {BesselJ}\left (2 i,2 \sqrt {2}\right )-2 \operatorname {BesselJ}\left (1+2 i,2 \sqrt {2}\right )\right ) \operatorname {BesselJ}\left (-2 i,2 \sqrt {2} \sqrt {t}\right )-\left (2 \operatorname {BesselJ}\left (-1-2 i,2 \sqrt {2}\right )+\sqrt {2} \operatorname {BesselJ}\left (-2 i,2 \sqrt {2}\right )-2 \operatorname {BesselJ}\left (1-2 i,2 \sqrt {2}\right )\right ) \operatorname {BesselJ}\left (2 i,2 \sqrt {2} \sqrt {t}\right )\right )}{\operatorname {BesselJ}\left (-1+2 i,2 \sqrt {2}\right ) \operatorname {BesselJ}\left (-2 i,2 \sqrt {2}\right )-\operatorname {BesselJ}\left (-1-2 i,2 \sqrt {2}\right ) \operatorname {BesselJ}\left (2 i,2 \sqrt {2}\right )+\operatorname {BesselJ}\left (2 i,2 \sqrt {2}\right ) \operatorname {BesselJ}\left (1-2 i,2 \sqrt {2}\right )-\operatorname {BesselJ}\left (-2 i,2 \sqrt {2}\right ) \operatorname {BesselJ}\left (1+2 i,2 \sqrt {2}\right )}
\]
✓ Sympy. Time used: 0.211 (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)
\[
x{\left (t \right )} = \frac {\sin {\left (\log {\left (t \right )} \right )} + 2 \cos {\left (\log {\left (t \right )} \right )}}{\sqrt {t}}
\]