Internal problem ID [1853]
Book: Differential equations and their applications, 4th ed., M. Braun
Section: Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number: 8.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=3 x_{1} \relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{1} \relax (t )+3 x_{2} \relax (t )\\ x_{3}^{\prime }\relax (t )&=3 x_{3} \relax (t )\\ x_{4}^{\prime }\relax (t )&=2 x_{3} \relax (t )+3 x_{4} \relax (t ) \end {align*}
With initial conditions \[ [x_{1} \relax (0) = 1, x_{2} \relax (0) = 1, x_{3} \relax (0) = 1, x_{4} \relax (0) = 1] \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 40
dsolve([diff(x__1(t),t) = 3*x__1(t), diff(x__2(t),t) = x__1(t)+3*x__2(t), diff(x__3(t),t) = 3*x__3(t), diff(x__4(t),t) = 2*x__3(t)+3*x__4(t), x__1(0) = 1, x__2(0) = 1, x__3(0) = 1, x__4(0) = 1],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1} \relax (t ) = {\mathrm e}^{3 t} \] \[ x_{2} \relax (t ) = {\mathrm e}^{3 t} \left (t +1\right ) \] \[ x_{3} \relax (t ) = {\mathrm e}^{3 t} \] \[ x_{4} \relax (t ) = {\mathrm e}^{3 t} \left (1+2 t \right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 44
DSolve[{x1'[t]==3*x1[t]+0*x2[t]+0*x3[t]+0*x4[t],x2'[t]==1*x1[t]+3*x2[t]-0*x3[t]+0*x4[t],x3'[t]==0*x1[t]-0*x2[t]+3*x3[t]-0*x4[t],x4'[t]==0*x1[t]-0*x2[t]+2*x3[t]+3*x4[t]},{x1[0]==1,x2[0]==1,x3[0]==1,x4[0]==1},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^{3 t} \\ \text {x2}(t)\to e^{3 t} (t+1) \\ \text {x3}(t)\to e^{3 t} \\ \text {x4}(t)\to e^{3 t} (2 t+1) \\ \end{align*}