Internal problem ID [331]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=4 x_{1} \left (t \right )+x_{2} \left (t \right )+4 x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+7 x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{3}^{\prime }\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: 0.031 (sec). Leaf size: 55
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}^{6 t}+c_{3} {\mathrm e}^{9 t} \\ x_{2} \left (t \right ) &= -2 c_{2} {\mathrm e}^{6 t}+c_{3} {\mathrm e}^{9 t} \\ x_{3} \left (t \right ) &= c_{2} {\mathrm e}^{6 t}+c_{3} {\mathrm e}^{9 t}-c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 158
DSolve[{x1'[t]==4*x1[t]+x2[t]+4*x3[t],x2'[t]==x1[t]+7*x2[t]+x3[t],x3'[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*}