65.10.1 problem 17.1

Internal problem ID [13705]
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 : 17.1
Date solved : Wednesday, March 05, 2025 at 10:13:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=t \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 12

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

\[ y = t \left (c_{1} +c_{2} {\mathrm e}^{t}\right ) \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 17

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

\[ y(t)\to e t \left (c_2 e^t+c_1\right ) \]

✓ Sympy. Time used: 0.774 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = C_{2} t^{2} \left (\frac {t^{3}}{24} + \frac {t^{2}}{6} + \frac {t}{2} + 1\right ) + C_{1} t + O\left (t^{6}\right ) \]