Internal problem ID [763]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Chapter 7.6, Complex Eigenvalues. page 417
Problem number: 23.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-\frac {x_{1} \relax (t )}{4}+x_{2} \relax (t )\\ x_{2}^{\prime }\relax (t )&=-x_{1} \relax (t )-\frac {x_{2} \relax (t )}{4}\\ x_{3}^{\prime }\relax (t )&=-\frac {x_{3} \relax (t )}{4} \end {align*}
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 47
dsolve([diff(x__1(t),t)=-1/4*x__1(t)+1*x__2(t)+0*x__3(t),diff(x__2(t),t)=-1*x__1(t)-1/4*x__2(t)+0*x__3(t),diff(x__3(t),t)=0*x__1(t)-0*x__2(t)-1/4*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = -{\mathrm e}^{-\frac {t}{4}} \left (\cos \relax (t ) c_{1}-\sin \relax (t ) c_{2}\right ) \] \[ x_{2} \relax (t ) = {\mathrm e}^{-\frac {t}{4}} \left (\sin \relax (t ) c_{1}+c_{2} \cos \relax (t )\right ) \] \[ x_{3} \relax (t ) = c_{3} {\mathrm e}^{-\frac {t}{4}} \]
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 110
DSolve[{x1'[t]==-1/4*x1[t]+1*x2[t]+0*x3[t],x2'[t]==-1*x1[t]-1/4*x2[t]+0*x3[t],x3'[t]==0*x1[t]-0*x2[t]-1/4*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^{-t/4} (c_1 \cos (t)+c_2 \sin (t)) \\ \text {x2}(t)\to e^{-t/4} (c_2 \cos (t)-c_1 \sin (t)) \\ \text {x3}(t)\to c_3 e^{-t/4} \\ \text {x1}(t)\to e^{-t/4} (c_1 \cos (t)+c_2 \sin (t)) \\ \text {x2}(t)\to e^{-t/4} (c_2 \cos (t)-c_1 \sin (t)) \\ \text {x3}(t)\to 0 \\ \end{align*}