7.25.30 problem 41

Internal problem ID [650]
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 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 41
Date solved : Monday, January 27, 2025 at 02:56:42 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )+7 x_{4} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+4 x_{2} \left (t \right )+10 x_{3} \left (t \right )+x_{4} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )+10 x_{2} \left (t \right )+4 x_{3} \left (t \right )+x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=7 x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )+4 x_{4} \left (t \right ) \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 61

dsolve([diff(x__1(t),t) = 4*x__1(t)+x__2(t)+x__3(t)+7*x__4(t), diff(x__2(t),t) = x__1(t)+4*x__2(t)+10*x__3(t)+x__4(t), diff(x__3(t),t) = x__1(t)+10*x__2(t)+4*x__3(t)+x__4(t), diff(x__4(t),t) = 7*x__1(t)+x__2(t)+x__3(t)+4*x__4(t), x__1(0) = 3, x__2(0) = 1, x__3(0) = 1, x__4(0) = 3], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{15 t}+2 \,{\mathrm e}^{10 t} \\ x_{2} \left (t \right ) &= 2 \,{\mathrm e}^{15 t}-{\mathrm e}^{10 t} \\ x_{3} \left (t \right ) &= 2 \,{\mathrm e}^{15 t}-{\mathrm e}^{10 t} \\ x_{4} \left (t \right ) &= {\mathrm e}^{15 t}+2 \,{\mathrm e}^{10 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 70

DSolve[{D[x1[t],t]==4*x1[t]+x2[t]+x3[t]+7*x4[t],D[x2[t],t]==x1[t]+4*x2[t]+10*x3[t]+x4[t],D[x3[t],t]==x1[t]+10*x2[t]+4*x3[t]+x4[t],D[x4[t],t]==7*x1[t]+x2[t]+x3[t]+4*x4[t]},{x1[0]==3,x2[0]==1,x3[0]==1,x4[0]==3},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{10 t} \left (e^{5 t}+2\right ) \\ \text {x2}(t)\to e^{10 t} \left (2 e^{5 t}-1\right ) \\ \text {x3}(t)\to e^{10 t} \left (2 e^{5 t}-1\right ) \\ \text {x4}(t)\to e^{10 t} \left (e^{5 t}+2\right ) \\ \end{align*}