57.11.7 problem 1(g)

Internal problem ID [14441]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.4.1 Cauchy-Euler equations. Exercises page 120
Problem number : 1(g)
Date solved : Thursday, October 02, 2025 at 09:37:15 AM
CAS classification : [[_2nd_order, _missing_y]]

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

With initial conditions

\begin{align*} x \left (1\right )&=0 \\ x^{\prime }\left (1\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 8

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

\[ x = 2 \ln \left (t \right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 9

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

\begin{align*} x(t)&\to 2 \log (t) \end{align*}

✓ Sympy. Time used: 0.077 (sec). Leaf size: 7

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

\[ x{\left (t \right )} = 2 \log {\left (t \right )} \]