57.11.6 problem 1(f)

Internal problem ID [14440]
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(f)
Date solved : Thursday, October 02, 2025 at 09:37:14 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} t^{2} x^{\prime \prime }+3 x^{\prime } t -8 x&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 14

ode:=t^2*diff(diff(x(t),t),t)+3*t*diff(x(t),t)-8*x(t) = 0; 
ic:=[x(1) = 0, D(x)(1) = 2]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {t^{6}-1}{3 t^{4}} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 17

ode=t^2*D[x[t],{t,2}]+3*t*D[x[t],t]-8*x[t]==0; 
ic={x[1]==0,Derivative[1][x][1 ]==2}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {t^6-1}{3 t^4} \end{align*}

✓ Sympy. Time used: 0.102 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(t**2*Derivative(x(t), (t, 2)) + 3*t*Derivative(x(t), t) - 8*x(t),0) 
ics = {x(1): 0, Subs(Derivative(x(t), t), t, 1): 2} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {t^{2}}{3} - \frac {1}{3 t^{4}} \]