9.4.17 problem problem 17

Internal problem ID [981]
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 17
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 )+4 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+7 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=4 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: 54

dsolve([diff(x__1(t),t)=4*x__1(t)+x__2(t)+4*x__3(t),diff(x__2(t),t)=x__1(t)+7*x__2(t)+x__3(t),diff(x__3(t),t)=4*x__1(t)+x__2(t)+4*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_1 +c_2 \,{\mathrm e}^{9 t}+c_3 \,{\mathrm e}^{6 t} \\ x_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{9 t}-2 c_3 \,{\mathrm e}^{6 t} \\ x_{3} \left (t \right ) &= c_2 \,{\mathrm e}^{9 t}+c_3 \,{\mathrm e}^{6 t}-c_1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 158

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

\begin{align*} \text {x1}(t)\to \frac {1}{6} \left (c_1 \left (e^{6 t}+2 e^{9 t}+3\right )+\left (e^{3 t}-1\right ) \left (3 c_3 e^{3 t}+2 (c_2+c_3) e^{6 t}+3 c_3\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{6 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}{6} \left (c_1 \left (e^{6 t}+2 e^{9 t}-3\right )+(c_3-2 c_2) e^{6 t}+2 (c_2+c_3) e^{9 t}+3 c_3\right ) \\ \end{align*}