74.12.54 problem 62

Internal problem ID [16361]
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 : Tuesday, January 28, 2025 at 09:06:56 AM
CAS classification : [_Lienard]

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

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 17

dsolve([t*diff(y(t),t$2)+2*diff(y(t),t)+t*y(t)=0,1/t*cos(t)],singsol=all)
 

\[ y = \frac {c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )}{t} \]

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 37

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

\[ y(t)\to \frac {2 c_1 e^{-i t}-i c_2 e^{i t}}{2 t} \]