14.9.5 problem 6

Internal problem ID [2571]
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 : 6
Date solved : Monday, January 27, 2025 at 06:01:00 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 6 y^{\prime \prime }+2 y^{\prime }+y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=1\\ y^{\prime }\left (2\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 0.252 (sec). Leaf size: 68

dsolve([6*diff(y(t),t$2)+2*diff(y(t),t)+y(t)=0,y(2) = 1, D(y)(2) = -1],y(t), singsol=all)
 

\[ y = -{\mathrm e}^{\frac {1}{3}-\frac {t}{6}} \left (\left (-\cos \left (\frac {\sqrt {5}}{3}\right )-\sin \left (\frac {\sqrt {5}}{3}\right ) \sqrt {5}\right ) \cos \left (\frac {t \sqrt {5}}{6}\right )+\sin \left (\frac {t \sqrt {5}}{6}\right ) \left (\cos \left (\frac {\sqrt {5}}{3}\right ) \sqrt {5}-\sin \left (\frac {\sqrt {5}}{3}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 51

DSolve[{6*D[y[t],{t,2}]+2*D[y[t],t]+y[t]==0,{y[2]==1,Derivative[1][y][2] ==-1}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{\frac {1}{3}-\frac {t}{6}} \left (\cos \left (\frac {1}{6} \sqrt {5} (t-2)\right )-\sqrt {5} \sin \left (\frac {1}{6} \sqrt {5} (t-2)\right )\right ) \]