Internal problem ID [377]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number: problem 20.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=2 x_{1} \relax (t )+x_{2} \relax (t )+x_{4} \relax (t )\\ x_{2}^{\prime }\relax (t )&=2 x_{2} \relax (t )+x_{3} \relax (t )\\ x_{3}^{\prime }\relax (t )&=2 x_{3} \relax (t )+x_{4} \relax (t )\\ x_{4}^{\prime }\relax (t )&=2 x_{4} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 78
dsolve([diff(x__1(t),t)=2*x__1(t)+1*x__2(t)+0*x__3(t)+1*x__4(t),diff(x__2(t),t)=0*x__1(t)+2*x__2(t)+1*x__3(t)+0*x__4(t),diff(x__3(t),t)=0*x__1(t)+0*x__2(t)+2*x__3(t)+1*x__4(t),diff(x__4(t),t)=0*x__1(t)+0*x__2(t)+0*x__3(t)+2*x__4(t)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1} \relax (t ) = \frac {\left (t^{3} c_{4}+3 t^{2} c_{3}+6 t c_{2}+6 t c_{4}+6 c_{1}\right ) {\mathrm e}^{2 t}}{6} \] \[ x_{2} \relax (t ) = \frac {\left (t^{2} c_{4}+2 t c_{3}+2 c_{2}\right ) {\mathrm e}^{2 t}}{2} \] \[ x_{3} \relax (t ) = \left (t c_{4}+c_{3}\right ) {\mathrm e}^{2 t} \] \[ x_{4} \relax (t ) = c_{4} {\mathrm e}^{2 t} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 94
DSolve[{x1'[t]==2*x1[t]+1*x2[t]+0*x3[t]+1*x4[t],x2'[t]==0*x1[t]+2*x2[t]+1*x3[t]+0*x4[t],x3'[t]==0*x1[t]+0*x2[t]+2*x3[t]+1*x4[t],x4'[t]==0*x1[t]+0*x2[t]+0*x3[t]+2*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{6} e^{2 t} \left (t \left (c_4 \left (t^2+6\right )+3 c_3 t+6 c_2\right )+6 c_1\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{2 t} (t (c_4 t+2 c_3)+2 c_2) \\ \text {x3}(t)\to e^{2 t} (c_4 t+c_3) \\ \text {x4}(t)\to c_4 e^{2 t} \\ \end{align*}