Internal problem ID [777]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 7.8, Repeated Eigenvalues. page 436
Problem number: 12.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-\frac {5 x_{1}\relax (t )}{2}+x_{2}\relax (t )+x_{3}\relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{1}\relax (t )-\frac {5 x_{2}\relax (t )}{2}+x_{3}\relax (t )\\ x_{3}^{\prime }\relax (t )&=x_{1}\relax (t )+x_{2}\relax (t )-\frac {5 x_{3}\relax (t )}{2} \end {align*}
With initial conditions \[ [x_{1}\relax (0) = 2, x_{2}\relax (0) = 3, x_{3}\relax (0) = -1] \]
✓ Solution by Maple
Time used: 0.158 (sec). Leaf size: 50
dsolve([diff(x__1(t),t) = -5/2*x__1(t)+x__2(t)+x__3(t), diff(x__2(t),t) = x__1(t)-5/2*x__2(t)+x__3(t), diff(x__3(t),t) = x__1(t)+x__2(t)-5/2*x__3(t), x__1(0) = 2, x__2(0) = 3, x__3(0) = -1],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1}\relax (t ) = \frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{3}+\frac {2 \,{\mathrm e}^{-\frac {7 t}{2}}}{3} \] \[ x_{2}\relax (t ) = \frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{3}+\frac {5 \,{\mathrm e}^{-\frac {7 t}{2}}}{3} \] \[ x_{3}\relax (t ) = \frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{3}-\frac {7 \,{\mathrm e}^{-\frac {7 t}{2}}}{3} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 71
DSolve[{x1'[t]==-5/2*x1[t]+1*x2[t]+1*x3[t],x2'[t]==1*x1[t]-5/2*x2[t]+1*x3[t],x3'[t]==1*x1[t]+1*x2[t]-5/2*x3[t]},{x1[0]==2,x2[0]==3,x3[0]==-1},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {2}{3} e^{-7 t/2} \left (2 e^{3 t}+1\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-7 t/2} \left (4 e^{3 t}+5\right ) \\ \text {x3}(t)\to \frac {1}{3} e^{-7 t/2} \left (4 e^{3 t}-7\right ) \\ \end{align*}