68.12.54 problem 62

Internal problem ID [17648]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 62
Date solved : Thursday, October 02, 2025 at 02:26:47 PM
CAS classification : [_Lienard]

\begin{align*} t y^{\prime \prime }+2 y^{\prime }+y t&=0 \end{align*}

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 37

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

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

✓ Sympy. Time used: 0.118 (sec). Leaf size: 20

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

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