8.9.5 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 : Tuesday, September 30, 2025 at 05:47:19 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*}

✓ Maple. Time used: 0.224 (sec). Leaf size: 68

ode:=6*diff(diff(y(t),t),t)+2*diff(y(t),t)+y(t) = 0; 
ic:=[y(2) = 1, D(y)(2) = -1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 51

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

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 0.219 (sec). Leaf size: 173

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) + 2*Derivative(y(t), t) + 6*Derivative(y(t), (t, 2)),0) 
ics = {y(2): 1, Subs(Derivative(y(t), t), t, 2): -1} 
dsolve(ode,func=y(t),ics=ics)
 

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