Internal problem ID [334]
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 20.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=5 x_{1}\relax (t )+x_{2}\relax (t )+3 x_{3}\relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{1}\relax (t )+7 x_{2}\relax (t )+x_{3}\relax (t )\\ x_{3}^{\prime }\relax (t )&=3 x_{1}\relax (t )+x_{2}\relax (t )+5 x_{3}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.023 (sec). Leaf size: 64
dsolve([diff(x__1(t),t)=5*x__1(t)+1*x__2(t)+3*x__3(t),diff(x__2(t),t)=1*x__1(t)+7*x__2(t)+1*x__3(t),diff(x__3(t),t)=3*x__1(t)+1*x__2(t)+5*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1}\relax (t ) = c_{1} {\mathrm e}^{6 t}-c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{9 t} \] \[ x_{2}\relax (t ) = -2 c_{1} {\mathrm e}^{6 t}+c_{3} {\mathrm e}^{9 t} \] \[ x_{3}\relax (t ) = c_{1} {\mathrm e}^{6 t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{9 t} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 141
DSolve[{x1'[t]==5*x1[t]+1*x2[t]+3*x3[t],x2'[t]==1*x1[t]+7*x2[t]+1*x3[t],x3'[t]==3*x1[t]+1*x2[t]+5*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{6} \left (3 (c_1-c_3) e^{2 t}+(c_1-2 c_2+c_3) e^{6 t}+2 (c_1+c_2+c_3) e^{9 t}\right ) \\ \text {x2}(t)\to \frac {1}{3} \left ((c_1+c_2+c_3) e^{9 t}-(c_1-2 c_2+c_3) e^{6 t}\right ) \\ \text {x3}(t)\to \frac {1}{6} \left (-3 (c_1-c_3) e^{2 t}+(c_1-2 c_2+c_3) e^{6 t}+2 (c_1+c_2+c_3) e^{9 t}\right ) \\ \end{align*}