14.9.13 problem 16

Internal problem ID [2579]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.2.2. Equal roots, reduction of order. Excercises page 149
Problem number : 16
Date solved : Monday, January 27, 2025 at 06:01:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 17

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

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

✓ Solution by Mathematica

Time used: 0.037 (sec). Leaf size: 39

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

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