7.24.20 problem 30 and 39

Internal problem ID [620]
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 : 30 and 39
Date solved : Monday, January 27, 2025 at 02:56:26 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )-4 x_{2} \left (t \right )-2 x_{4} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{2} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=6 x_{1} \left (t \right )-12 x_{2} \left (t \right )-x_{3} \left (t \right )-6 x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=-4 x_{2} \left (t \right )-x_{4} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = 1\\ x_{3} \left (0\right ) = 1\\ x_{4} \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.063 (sec). Leaf size: 48

dsolve([diff(x__1(t),t) = x__1(t)-4*x__2(t)-2*x__4(t), diff(x__2(t),t) = x__2(t), diff(x__3(t),t) = 6*x__1(t)-12*x__2(t)-x__3(t)-6*x__4(t), diff(x__4(t),t) = -4*x__2(t)-x__4(t), x__1(0) = 1, x__2(0) = 1, x__3(0) = 1, x__4(0) = 1], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= -2 \,{\mathrm e}^{t}+3 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} \\ x_{3} \left (t \right ) &= -6 \,{\mathrm e}^{t}+7 \,{\mathrm e}^{-t} \\ x_{4} \left (t \right ) &= -2 \,{\mathrm e}^{t}+3 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 56

DSolve[{D[x1[t],t]==x1[t]-4*x2[t]-2*x4[t],D[x2[t],t]==x2[t],D[x3[t],t]==6*x1[t]-12*x2[t]-x3[t]-6*x4[t],D[x4[t],t]==-4*x2[t]-x4[t]},{x1[0]==1,x2[0]==1,x3[0]==1,x4[0]==1},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to 3 e^{-t}-2 e^t \\ \text {x2}(t)\to e^t \\ \text {x3}(t)\to 7 e^{-t}-6 e^t \\ \text {x4}(t)\to 3 e^{-t}-2 e^t \\ \end{align*}