Internal problem ID [387]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number: problem 30.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right )+x_{4} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=3 x_{2} \left (t \right )-5 x_{3} \left (t \right )+3 x_{4} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=-13 x_{2} \left (t \right )+22 x_{3} \left (t \right )-12 x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=-27 x_{2} \left (t \right )+45 x_{3} \left (t \right )-25 x_{4} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 89
dsolve([diff(x__1(t),t)=2*x__1(t)+1*x__2(t)-2*x__3(t)+1*x__4(t),diff(x__2(t),t)=0*x__1(t)+3*x__2(t)-5*x__3(t)+3*x__4(t),diff(x__3(t),t)=0*x__1(t)-13*x__2(t)+22*x__3(t)-12*x__4(t),diff(x__4(t),t)=0*x__1(t)-27*x__2(t)+45*x__3(t)-25*x__4(t)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= \frac {\left (-c_{2} t +3 c_{1} \right ) {\mathrm e}^{2 t}}{3} \\ x_{2} \left (t \right ) &= {\mathrm e}^{-t} \left (c_{4} t +c_{3} \right ) \\ x_{3} \left (t \right ) &= \left (-{\mathrm e}^{-3 t} \left (c_{4} t +c_{3} -c_{4} \right )+c_{2} \right ) {\mathrm e}^{2 t} \\ x_{4} \left (t \right ) &= -3 c_{3} {\mathrm e}^{-t}-3 c_{4} {\mathrm e}^{-t} t +2 c_{4} {\mathrm e}^{-t}+\frac {5 c_{2} {\mathrm e}^{2 t}}{3} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 161
DSolve[{x1'[t]==2*x1[t]+1*x2[t]-2*x3[t]+1*x4[t],x2'[t]==0*x1[t]+3*x2[t]-5*x3[t]+3*x4[t],x3'[t]==0*x1[t]-13*x2[t]+22*x3[t]-12*x4[t],x4'[t]==0*x1[t]-27*x2[t]+45*x3[t]-25*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^{2 t} ((c_2-2 c_3+c_4) t+c_1) \\ \text {x2}(t)\to e^{-t} (4 c_2 t-5 c_3 t+3 c_4 t+c_2) \\ \text {x3}(t)\to e^{-t} \left (c_2 \left (-4 t-3 e^{3 t}+3\right )+c_3 \left (5 t+6 e^{3 t}-5\right )-3 c_4 \left (t+e^{3 t}-1\right )\right ) \\ \text {x4}(t)\to e^{-t} \left (c_2 \left (-12 t-5 e^{3 t}+5\right )+5 c_3 \left (3 t+2 e^{3 t}-2\right )-c_4 \left (9 t+5 e^{3 t}-6\right )\right ) \\ \end{align*}