14.23.7 problem 7

Internal problem ID [2746]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number : 7
Date solved : Monday, January 27, 2025 at 06:12:39 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = -1\\ x_{3} \left (0\right ) = -2 \end{align*}

✓ Solution by Maple

Time used: 0.054 (sec). Leaf size: 94

dsolve([diff(x__1(t),t) = -3*x__1(t)+2*x__3(t), diff(x__2(t),t) = x__1(t)-x__2(t), diff(x__3(t),t) = -2*x__1(t)-x__2(t), x__1(0) = 0, x__2(0) = -1, x__3(0) = -2], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= 2 \,{\mathrm e}^{-2 t}-\sqrt {2}\, {\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right )-2 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right ) \\ x_{2} \left (t \right ) &= -2 \,{\mathrm e}^{-2 t}+{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right )-\sqrt {2}\, {\mathrm e}^{-t} \sin \left (\sqrt {2}\, t \right ) \\ x_{3} \left (t \right ) &= {\mathrm e}^{-2 t}-3 \,{\mathrm e}^{-t} \cos \left (\sqrt {2}\, t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.042 (sec). Leaf size: 109

DSolve[{D[ x1[t],t]==-3*x1[t]-0*x2[t]+2*x3[t],D[ x2[t],t]==1*x1[t]-1*x2[t]-0*x3[t],D[ x3[t],t]==-2*x1[t]-1*x2[t]-0*x3[t]},{x1[0]==0,x2[0]==-1,x3[0]==-2},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to -e^{-2 t} \left (\sqrt {2} e^t \sin \left (\sqrt {2} t\right )+2 e^t \cos \left (\sqrt {2} t\right )-2\right ) \\ \text {x2}(t)\to e^{-2 t} \left (-\sqrt {2} e^t \sin \left (\sqrt {2} t\right )+e^t \cos \left (\sqrt {2} t\right )-2\right ) \\ \text {x3}(t)\to e^{-2 t} \left (1-3 e^t \cos \left (\sqrt {2} t\right )\right ) \\ \end{align*}