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: 0.031 (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)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1} \left (t \right ) = \frac {c_{1} {\mathrm e}^{t}}{3}+c_{4} {\mathrm e}^{-t} \] \[ x_{2} \left (t \right ) = -\frac {c_{3} {\mathrm e}^{t}}{2} \] \[ x_{3} \left (t \right ) = c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t} \] \[ x_{4} \left (t \right ) = c_{3} {\mathrm e}^{t}+c_{4} {\mathrm e}^{-t} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 81
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 c_1 \cosh (t)+(c_1-2 (2 c_2+c_4)) \sinh (t) \\ \text {x2}(t)\to c_2 e^t \\ \text {x3}(t)\to c_3 \cosh (t)-(-6 c_1+12 c_2+c_3+6 c_4) \sinh (t) \\ \text {x4}(t)\to c_4 e^{-t}-4 c_2 \sinh (t) \\ \end{align*}