Internal problem ID [1849]
Book: Differential equations and their applications, 4th ed., M. Braun
Section: Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number: 4.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=2 x_{1}\relax (t )-x_{3}\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}^{\prime }\relax (t )&=-x_{3}\relax (t )+2 x_{4}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.08 (sec). Leaf size: 56
dsolve([diff(x__1(t),t)=2*x__1(t)+0*x__2(t)-1*x__3(t)+0*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)-0*x__4(t),diff(x__4(t),t)=0*x__1(t)-0*x__2(t)-1*x__3(t)+2*x__4(t)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1}\relax (t ) = {\mathrm e}^{2 t} \left (c_{4} t +c_{1}+c_{3}\right ) \] \[ x_{2}\relax (t ) = {\mathrm e}^{2 t} \left (-c_{4} t +c_{2}-c_{3}\right ) \] \[ x_{3}\relax (t ) = -c_{4} {\mathrm e}^{2 t} \] \[ x_{4}\relax (t ) = {\mathrm e}^{2 t} \left (c_{4} t +c_{3}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 63
DSolve[{x1'[t]==2*x1[t]+0*x2[t]-1*x3[t]+0*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]-0*x4[t],x4'[t]==0*x1[t]-0*x2[t]-1*x3[t]+2*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^{2 t} (c_1-c_3 t) \\ \text {x2}(t)\to e^{2 t} (c_3 t+c_2) \\ \text {x3}(t)\to c_3 e^{2 t} \\ \text {x4}(t)\to e^{2 t} (c_4-c_3 t) \\ \end{align*}