65.12.10 problem 19.1 (x)

Internal problem ID [13805]
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 : Tuesday, January 28, 2025 at 06:04:10 AM
CAS classification : [[_Emden, _Fowler]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.108 (sec). Leaf size: 32

dsolve([t^2*diff(x(t),t$2)+3*t*diff(x(t),t)+13*x(t)=0,x(1) = -1, D(x)(1) = 2],x(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 41

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

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