90.24.8 problem 8

Internal problem ID [25373]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 5. Second Order Linear Differential Equations. Exercises at page 371
Problem number : 8
Date solved : Friday, October 03, 2025 at 12:00:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t y^{\prime \prime }-2 \left (t +1\right ) y^{\prime }+4 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{2 t} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 20

ode:=t*diff(diff(y(t),t),t)-2*(t+1)*diff(y(t),t)+4*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{2 t}+2 c_2 \left (t^{2}+t +\frac {1}{2}\right ) \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 30

ode=t*D[y[t],{t,2}]-2*(t+1)*D[y[t],t]+4*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to c_1 e^{2 t}-\frac {1}{4} c_2 \left (2 t^2+2 t+1\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*Derivative(y(t), (t, 2)) - (2*t + 2)*Derivative(y(t), t) + 4*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

False