9.4.19 problem problem 19

Internal problem ID [983]
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 19
Date solved : Monday, January 27, 2025 at 03:22:43 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 )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+4 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+4 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 63

dsolve([diff(x__1(t),t)=4*x__1(t)+1*x__2(t)+1*x__3(t),diff(x__2(t),t)=1*x__1(t)+4*x__2(t)+1*x__3(t),diff(x__3(t),t)=1*x__1(t)+1*x__2(t)+4*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{3 t}+c_3 \,{\mathrm e}^{6 t} \\ x_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{3 t}+c_3 \,{\mathrm e}^{6 t}+c_1 \,{\mathrm e}^{3 t} \\ x_{3} \left (t \right ) &= -2 c_2 \,{\mathrm e}^{3 t}+c_3 \,{\mathrm e}^{6 t}-c_1 \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 124

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

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{3 t} \left (c_1 \left (e^{3 t}+2\right )+(c_2+c_3) \left (e^{3 t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{3 t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}+2\right )+c_3 \left (e^{3 t}-1\right )\right ) \\ \text {x3}(t)\to \frac {1}{3} e^{3 t} \left (c_1 \left (e^{3 t}-1\right )+c_2 \left (e^{3 t}-1\right )+c_3 \left (e^{3 t}+2\right )\right ) \\ \end{align*}