9.4.30 problem problem 41

Internal problem ID [994]
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 41
Date solved : Monday, January 27, 2025 at 03:22:47 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \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 )\\ \frac {d}{d t}x_{2} \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 )\\ \frac {d}{d t}x_{3} \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 )\\ \frac {d}{d t}x_{4} \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: 0.018 (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: 0.009 (sec). Leaf size: 70

DSolve[{D[ x1[t],t]==4*x1[t]+1*x2[t]+1*x3[t]+7*x4[t],D[ x2[t],t]==1*x1[t]+4*x2[t]+10*x3[t]+1*x4[t],D[ x3[t],t]==1*x1[t]+10*x2[t]+4*x3[t]+1*x4[t],D[ x4[t],t]==7*x1[t]+1*x2[t]+1*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*}