90.24.14 problem 14

Internal problem ID [25379]
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 : 14
Date solved : Friday, October 03, 2025 at 12:00:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=t \cos \left (t \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 15

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

\[ y = t \left (c_1 \sin \left (t \right )+c_2 \cos \left (t \right )\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 33

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

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

✓ Sympy. Time used: 0.136 (sec). Leaf size: 20

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

\[ y{\left (t \right )} = t^{\frac {3}{2}} \left (C_{1} J_{\frac {1}{2}}\left (t\right ) + C_{2} Y_{\frac {1}{2}}\left (t\right )\right ) \]