59.12.4 problem 19.1 (iv)

Internal problem ID [15080]
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 : Thursday, October 02, 2025 at 10:02:44 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 5

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

\[ x = t \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 172

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

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

✓ Sympy. Time used: 0.090 (sec). Leaf size: 3

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

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