Internal problem ID [338]
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 24.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=2 x_{1} \relax (t )+x_{2} \relax (t )-x_{3} \relax (t )\\ x_{2}^{\prime }\relax (t )&=-4 x_{1} \relax (t )-3 x_{2} \relax (t )-x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=4 x_{1} \relax (t )+4 x_{2} \relax (t )+2 x_{3} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 77
dsolve([diff(x__1(t),t)=2*x__1(t)+1*x__2(t)-1*x__3(t),diff(x__2(t),t)=-4*x__1(t)-3*x__2(t)-1*x__3(t),diff(x__3(t),t)=4*x__1(t)+4*x__2(t)+2*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1} \relax (t ) = \frac {c_{2} \cos \left (2 t \right )}{2}-\frac {c_{3} \sin \left (2 t \right )}{2}+\frac {c_{2} \sin \left (2 t \right )}{2}+\frac {c_{3} \cos \left (2 t \right )}{2}-c_{1} {\mathrm e}^{t} \] \[ x_{2} \relax (t ) = -c_{2} \sin \left (2 t \right )-c_{3} \cos \left (2 t \right )+c_{1} {\mathrm e}^{t} \] \[ x_{3} \relax (t ) = c_{2} \sin \left (2 t \right )+c_{3} \cos \left (2 t \right ) \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 101
DSolve[{x1'[t]==2*x1[t]+1*x2[t]-1*x3[t],x2'[t]==-4*x1[t]-3*x2[t]-1*x3[t],x3'[t]==4*x1[t]+4*x2[t]+2*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to (c_2+c_3) \left (-e^t\right )+(c_1+c_2+c_3) \cos (2 t)+(c_1+c_2) \sin (2 t) \\ \text {x2}(t)\to (c_2+c_3) e^t-c_3 \cos (2 t)-(2 (c_1+c_2)+c_3) \sin (2 t) \\ \text {x3}(t)\to c_3 \cos (2 t)+(2 (c_1+c_2)+c_3) \sin (2 t) \\ \end{align*}