13.9.6 problem 7

Internal problem ID [2392]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.2, Equal roots, reduction of order. Page 147
Problem number : 7
Date solved : Monday, January 27, 2025 at 05:51:25 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 9 y^{\prime \prime }-12 y^{\prime }+4 y&=0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 19

dsolve([9*diff(y(t),t$2)-12*diff(y(t),t)+4*y(t)=0,y(Pi) = 0, D(y)(Pi) = 2],y(t), singsol=all)
 

\[ y = -2 \,{\mathrm e}^{-\frac {2 \pi }{3}+\frac {2 t}{3}} \left (-t +\pi \right ) \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 24

DSolve[{9*D[y[t],{t,2}]-12*D[y[t],t]+4*y[t]==0,{y[Pi]==0,Derivative[1][y][Pi]==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-\frac {2}{3} (\pi -t)} (2 t-2 \pi ) \]