9.4.26 problem problem 26

Internal problem ID [990]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 26
Date solved : Wednesday, February 05, 2025 at 04:51:29 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=9 x_{1} \left (t \right )-x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-9 x_{1} \left (t \right )+4 x_{2} \left (t \right )-x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 0\\ x_{3} \left (0\right ) = 17 \end{align*}

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 63

dsolve([diff(x__1(t),t) = 3*x__1(t)+x__3(t), diff(x__2(t),t) = 9*x__1(t)-x__2(t)+2*x__3(t), diff(x__3(t),t) = -9*x__1(t)+4*x__2(t)-x__3(t), x__1(0) = 0, x__2(0) = 0, x__3(0) = 17], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= 4 \,{\mathrm e}^{3 t}+{\mathrm e}^{-t} \sin \left (t \right )-4 \,{\mathrm e}^{-t} \cos \left (t \right ) \\ x_{2} \left (t \right ) &= 9 \,{\mathrm e}^{3 t}-9 \,{\mathrm e}^{-t} \cos \left (t \right )-2 \,{\mathrm e}^{-t} \sin \left (t \right ) \\ x_{3} \left (t \right ) &= 17 \,{\mathrm e}^{-t} \cos \left (t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 62

DSolve[{D[ x1[t],t]==3*x1[t]+0*x2[t]+1*x3[t],D[ x2[t],t]==9*x1[t]-1*x2[t]+2*x3[t],D[ x3[t],t]==-9*x1[t]+4*x2[t]-1*x3[t]},{x1[0]==0,x2[0]==0,x3[0]==17},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-t} \left (4 e^{4 t}+\sin (t)-4 \cos (t)\right ) \\ \text {x2}(t)\to e^{-t} \left (9 e^{4 t}-2 \sin (t)-9 \cos (t)\right ) \\ \text {x3}(t)\to 17 e^{-t} \cos (t) \\ \end{align*}