13.8.6 problem 5

Internal problem ID [2381]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2.1, Complex roots. Page 141
Problem number : 5
Date solved : Monday, January 27, 2025 at 05:50:55 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: 0.034 (sec). Leaf size: 32

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)
 

\[ y = \frac {{\mathrm e}^{-\frac {t}{2}} \left (5 \sqrt {7}\, \sin \left (\frac {\sqrt {7}\, t}{2}\right )+7 \cos \left (\frac {\sqrt {7}\, t}{2}\right )\right )}{7} \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 48

DSolve[{2*D[y[t],{t,2}]+3*D[y[t],t]+4*y[t]==0,{y[0]==1,Derivative[1][y][0] ==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{23} e^{-3 t/4} \left (11 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{4}\right )+23 \cos \left (\frac {\sqrt {23} t}{4}\right )\right ) \]