Internal problem ID [378]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number: problem 21.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-x_{1}\relax (t )-4 x_{2}\relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{1}\relax (t )+3 x_{2}\relax (t )\\ x_{3}^{\prime }\relax (t )&=x_{1}\relax (t )+2 x_{2}\relax (t )+x_{3}\relax (t )\\ x_{4}^{\prime }\relax (t )&=x_{2}\relax (t )+x_{4}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.1 (sec). Leaf size: 59
dsolve([diff(x__1(t),t)=-1*x__1(t)-4*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__2(t),t)=1*x__1(t)+3*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__3(t),t)=1*x__1(t)+2*x__2(t)+1*x__3(t)+0*x__4(t),diff(x__4(t),t)=0*x__1(t)+1*x__2(t)+0*x__3(t)+1*x__4(t)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1}\relax (t ) = -2 \,{\mathrm e}^{t} \left (2 c_{4} t +c_{3}-c_{4}\right ) \] \[ x_{2}\relax (t ) = {\mathrm e}^{t} \left (2 c_{4} t +c_{3}\right ) \] \[ x_{3}\relax (t ) = {\mathrm e}^{t} \left (2 c_{4} t +c_{1}+c_{3}\right ) \] \[ x_{4}\relax (t ) = {\mathrm e}^{t} \left (c_{4} t^{2}+c_{3} t +c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 89
DSolve[{x1'[t]==-1*x1[t]-4*x2[t]+0*x3[t]+0*x4[t],x2'[t]==1*x1[t]+3*x2[t]+0*x3[t]+0*x4[t],x3'[t]==1*x1[t]+2*x2[t]+1*x3[t]+0*x4[t],x4'[t]==0*x1[t]+1*x2[t]+0*x3[t]+1*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^t (-2 c_1 t-4 c_2 t+c_1) \\ \text {x2}(t)\to e^t ((c_1+2 c_2) t+c_2) \\ \text {x3}(t)\to e^t ((c_1+2 c_2) t+c_3) \\ \text {x4}(t)\to \frac {1}{2} e^t \left (c_1 t^2+2 c_2 (t+1) t+2 c_4\right ) \\ \end{align*}