7.24.16 problem 26 and 35

Internal problem ID [616]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.3 (Matrices and linear systems). Problems at page 364
Problem number : 26 and 35
Date solved : Monday, January 27, 2025 at 02:56:23 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 51

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

\begin{align*} x_{1} \left (t \right ) &= 2 \,{\mathrm e}^{5 t}-4 \,{\mathrm e}^{3 t}+2 \,{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= -2 \,{\mathrm e}^{5 t}+2 \,{\mathrm e}^{t} \\ x_{3} \left (t \right ) &= {\mathrm e}^{5 t}+2 \,{\mathrm e}^{3 t}+{\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 50

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

\begin{align*} \text {x1}(t)\to 2 e^t \left (e^{2 t}-1\right )^2 \\ \text {x2}(t)\to -2 e^t \left (e^{4 t}-1\right ) \\ \text {x3}(t)\to e^t \left (e^{2 t}+1\right )^2 \\ \end{align*}