74.12.52 problem 61

Internal problem ID [16359]
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 : 61
Date solved : Tuesday, January 28, 2025 at 09:06:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} y^{\prime \prime }-4 y^{\prime } t +\left (t^{2}+6\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=t^{2} \cos \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 17

dsolve([t^2*diff(y(t),t$2)-4*t*diff(y(t),t)+(t^2+6)*y(t)=0,t^2*cos(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.030 (sec). Leaf size: 37

DSolve[t^2*D[y[t],{t,2}]-4*t*D[y[t],t]+(t^2+6)*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{2} e^{-i t} t^2 \left (2 c_1-i c_2 e^{2 i t}\right ) \]