65.12.4 problem 19.1 (iv)

Internal problem ID [13799]
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 (iv)
Date solved : Tuesday, January 28, 2025 at 06:03:52 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 5

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

\[ x \left (t \right ) = t \]

✓ Solution by Mathematica

Time used: 0.073 (sec). Leaf size: 172

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

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