59.10.4 problem 17.4

Internal problem ID [15068]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 17, Reduction of order. Exercises page 162
Problem number : 17.4
Date solved : Thursday, October 02, 2025 at 10:02:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} x&={\mathrm e}^{-t} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 16

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

\[ x = \frac {c_1}{t}+c_2 \,{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.046 (sec). Leaf size: 42

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

\begin{align*} x(t)&\to \frac {e^{-t} \sqrt {1-t} \left (c_1 e^t-c_2 t\right )}{\sqrt {t-1} t} \end{align*}

✗ Sympy

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

False