58.22.4 problem 1 (d)

Internal problem ID [14950]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 11, The nth order homogeneous linear differential equation. Section 11.8, Exercises page 583
Problem number : 1 (d)
Date solved : Thursday, October 02, 2025 at 09:56:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{3} x^{\prime \prime }-\left (t^{3}+2 t^{2}-t \right ) x^{\prime }+\left (t^{2}+t -1\right ) x&=0 \end{align*}

✓ Maple. Time used: 0.063 (sec). Leaf size: 47

ode:=t^3*diff(diff(x(t),t),t)-(t^3+2*t^2-t)*diff(x(t),t)+(t^2+t-1)*x(t) = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = \left ({\mathrm e}^{t} \operatorname {HeunD}\left (-4, -5, 8, -3, \frac {t -1}{t +1}\right ) c_2 +{\mathrm e}^{\frac {1}{t}} \operatorname {HeunD}\left (4, -5, 8, -3, \frac {t -1}{t +1}\right ) c_1 \right ) t^{{3}/{2}} \]

✓ Mathematica. Time used: 0.17 (sec). Leaf size: 34

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

\begin{align*} x(t)&\to e^t t \left (c_2 \int _1^te^{\frac {1}{K[1]}-K[1]}dK[1]+c_1\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(x(t), t) - (t**3*Derivative(x(t), (t, 2)) + t**2*x(t) + t*x(t) - x(t))/(t*(t**2 + 2*t - 1)) cannot be solved by the factorable group method