14.8.5 problem 5

Internal problem ID [2560]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.2.1 Linear equations with constant coefficients (complex roots). Excercises page 144
Problem number : 5
Date solved : Monday, January 27, 2025 at 06:00:30 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-2 \end{align*}

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 29

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

\[ y = {\mathrm e}^{-\frac {t}{2}} \left (-\frac {3 \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right )}{7}+\cos \left (\frac {\sqrt {7}\, t}{2}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 48

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

\[ y(t)\to \frac {1}{7} e^{-t/2} \left (7 \cos \left (\frac {\sqrt {7} t}{2}\right )-3 \sqrt {7} \sin \left (\frac {\sqrt {7} t}{2}\right )\right ) \]