Internal problem ID [333]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=4 x_{1} \relax (t )+x_{2} \relax (t )+x_{3} \relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{1} \relax (t )+4 x_{2} \relax (t )+x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=x_{1} \relax (t )+x_{2} \relax (t )+4 x_{3} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 64
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)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = c_{2} {\mathrm e}^{6 t}-2 c_{3} {\mathrm e}^{3 t}-c_{1} {\mathrm e}^{3 t} \] \[ x_{2} \relax (t ) = c_{2} {\mathrm e}^{6 t}+c_{3} {\mathrm e}^{3 t}+c_{1} {\mathrm e}^{3 t} \] \[ x_{3} \relax (t ) = c_{2} {\mathrm e}^{6 t}+c_{3} {\mathrm e}^{3 t} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 112
DSolve[{x1'[t]==4*x1[t]+1*x2[t]+1*x3[t],x2'[t]==1*x1[t]+4*x2[t]+1*x3[t],x3'[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} \left ((c_1+c_2+c_3) e^{6 t}-(c_1-2 c_2+c_3) e^{3 t}\right ) \\ \text {x3}(t)\to \frac {1}{3} \left ((c_1+c_2+c_3) e^{6 t}-(c_1+c_2-2 c_3) e^{3 t}\right ) \\ \end{align*}