74.9.27 problem 45

Internal problem ID [16203]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.1, page 141
Problem number : 45
Date solved : Tuesday, January 28, 2025 at 08:57:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t y^{\prime \prime }+2 y^{\prime }+16 t y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {\sin \left (4 t \right )}{t} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 21

dsolve([t*diff(y(t),t$2)+2*diff(y(t),t)+16*t*y(t)=0,1/t*sin(4*t)],singsol=all)
 

\[ y = \frac {c_{1} \sin \left (4 t \right )+c_{2} \cos \left (4 t \right )}{t} \]

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 37

DSolve[t*D[y[t],{t,2}]+2*D[y[t],t]+16*t*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {8 c_1 e^{-4 i t}-i c_2 e^{4 i t}}{8 t} \]