Internal problem ID [376]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number: problem 19.
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 )-2 x_{4} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{2} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=6 x_{1} \left (t \right )-12 x_{2} \left (t \right )-x_{3} \left (t \right )-6 x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=-4 x_{2} \left (t \right )-x_{4} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 53
dsolve([diff(x__1(t),t)=1*x__1(t)-4*x__2(t)+0*x__3(t)-2*x__4(t),diff(x__2(t),t)=0*x__1(t)+1*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__3(t),t)=6*x__1(t)-12*x__2(t)-1*x__3(t)-6*x__4(t),diff(x__4(t),t)=0*x__1(t)-4*x__2(t)+0*x__3(t)-1*x__4(t)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= c_{2} {\mathrm e}^{t}+c_{3} {\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= c_{4} {\mathrm e}^{t} \\ x_{3} \left (t \right ) &= 3 c_{2} {\mathrm e}^{t}+{\mathrm e}^{-t} c_{1} \\ x_{4} \left (t \right ) &= -2 c_{4} {\mathrm e}^{t}+c_{3} {\mathrm e}^{-t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 114
DSolve[{x1'[t]==1*x1[t]-4*x2[t]+0*x3[t]-2*x4[t],x2'[t]==0*x1[t]+1*x2[t]+0*x3[t]+0*x4[t],x3'[t]==6*x1[t]-12*x2[t]-1*x3[t]-6*x4[t],x4'[t]==0*x1[t]-4*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} \left ((c_1-2 c_2-c_4) e^{2 t}+2 c_2+c_4\right ) \\ \text {x2}(t)\to c_2 e^t \\ \text {x3}(t)\to e^{-t} \left (3 c_1 \left (e^{2 t}-1\right )-6 c_2 \left (e^{2 t}-1\right )-3 c_4 e^{2 t}+c_3+3 c_4\right ) \\ \text {x4}(t)\to e^{-t} \left (c_4-2 c_2 \left (e^{2 t}-1\right )\right ) \\ \end{align*}