10.16.10 problem 10

Internal problem ID [1410]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 10
Date solved : Monday, January 27, 2025 at 04:56:55 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{1} \left (t \right )+2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )-x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = -2 \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 34

dsolve([diff(x__1(t),t) = -3*x__1(t)+2*x__2(t), diff(x__2(t),t) = -x__1(t)-x__2(t), x__1(0) = 1, x__2(0) = -2], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-2 t} \left (-5 \sin \left (t \right )+\cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-2 t} \left (-4 \cos \left (t \right )-6 \sin \left (t \right )\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 27

DSolve[{D[ x1[t],t]==-3*x1[t]+2*x2[t],D[ x2[t],t]==-1*x1[t]-1*x2[t]},{x1[0]==1,x2[0]==1},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-2 t} (\sin (t)+\cos (t)) \\ \text {x2}(t)\to e^{-2 t} \cos (t) \\ \end{align*}