76.12.20 problem 32

Internal problem ID [17576]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.2 (Theory of second order linear homogeneous equations). Problems at page 226
Problem number : 32
Date solved : Tuesday, January 28, 2025 at 10:44:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=t \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 12

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

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

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 16

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

\[ y(t)\to t \left (c_2 e^t+c_1\right ) \]