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 }\left (t \right )&=-x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+3 x_{2} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=x_{2} \left (t \right )+x_{4} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 63
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)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= -{\mathrm e}^{t} \left (2 c_{4} t +2 c_{3} -c_{4} \right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} \left (c_{4} t +c_{3} \right ) \\ x_{3} \left (t \right ) &= {\mathrm e}^{t} \left (c_{4} t +c_{1} +c_{3} \right ) \\ x_{4} \left (t \right ) &= \frac {\left (c_{4} t^{2}+2 c_{3} t +2 c_{2} \right ) {\mathrm e}^{t}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 91
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 (c_1 (2 t-1)+4 c_2 t) \\ \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*}