59.12.10 problem 19.1 (x)

Internal problem ID [15086]
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 (x)
Date solved : Thursday, October 02, 2025 at 10:02:52 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t^{2} x^{\prime \prime }+3 x^{\prime } t +13 x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.117 (sec). Leaf size: 32

ode:=t^2*diff(diff(x(t),t),t)+3*t*diff(x(t),t)+13*x(t) = 0; 
ic:=[x(1) = -1, D(x)(1) = 2]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {\sqrt {3}\, \sin \left (2 \sqrt {3}\, \ln \left (t \right )\right )-6 \cos \left (2 \sqrt {3}\, \ln \left (t \right )\right )}{6 t} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 41

ode=t^2*D[x[t],{t,2}]+3*t*D[x[t],t]+13*x[t]==0; 
ic={x[1]==-1,Derivative[1][x][1 ]==2}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {\sqrt {3} \sin \left (2 \sqrt {3} \log (t)\right )-6 \cos \left (2 \sqrt {3} \log (t)\right )}{6 t} \end{align*}

✓ Sympy. Time used: 0.134 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t**2*Derivative(x(t), (t, 2)) + 3*t*Derivative(x(t), t) + 13*x(t),0) 
ics = {x(1): -1, Subs(Derivative(x(t), t), t, 1): 2} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {\frac {\sqrt {3} \sin {\left (2 \sqrt {3} \log {\left (t \right )} \right )}}{6} - \cos {\left (2 \sqrt {3} \log {\left (t \right )} \right )}}{t} \]