Internal problem ID [349]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number: problem 46.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=13 x_{1} \left (t \right )-42 x_{2} \left (t \right )+106 x_{3} \left (t \right )+139 x_{4} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-16 x_{2} \left (t \right )+52 x_{3} \left (t \right )+70 x_{4} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )+6 x_{2} \left (t \right )-20 x_{3} \left (t \right )-31 x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=-x_{1} \left (t \right )-6 x_{2} \left (t \right )+22 x_{3} \left (t \right )+33 x_{4} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 123
dsolve([diff(x__1(t),t)=13*x__1(t)-42*x__2(t)+106*x__3(t)+139*x__4(t),diff(x__2(t),t)=2*x__1(t)-16*x__2(t)+52*x__3(t)+70*x__4(t),diff(x__3(t),t)=1*x__1(t)+6*x__2(t)-20*x__3(t)-31*x__4(t),diff(x__4(t),t)=-1*x__1(t)-6*x__2(t)+22*x__3(t)+33*x__4(t)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{4 t}+c_{2} {\mathrm e}^{-4 t}+c_{3} {\mathrm e}^{2 t}+c_{4} {\mathrm e}^{8 t} \\ x_{2} \left (t \right ) &= c_{1} {\mathrm e}^{4 t}+\frac {2 c_{2} {\mathrm e}^{-4 t}}{3}+2 c_{3} {\mathrm e}^{2 t}-\frac {2 c_{4} {\mathrm e}^{8 t}}{3} \\ x_{3} \left (t \right ) &= -c_{1} {\mathrm e}^{4 t}-\frac {c_{2} {\mathrm e}^{-4 t}}{3}+2 c_{3} {\mathrm e}^{2 t}+c_{4} {\mathrm e}^{8 t} \\ x_{4} \left (t \right ) &= c_{1} {\mathrm e}^{4 t}+\frac {c_{2} {\mathrm e}^{-4 t}}{3}-c_{3} {\mathrm e}^{2 t}-c_{4} {\mathrm e}^{8 t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 449
DSolve[{x1'[t]==13*x1[t]-42*x2[t]+106*x3[t]+139*x4[t],x2'[t]==2*x1[t]-16*x2[t]+52*x3[t]+70*x4[t],x3'[t]==1*x1[t]+6*x2[t]-20*x3[t]-31*x4[t],x4'[t]==-1*x1[t]-6*x2[t]+22*x3[t]+33*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-4 t} \left (c_1 \left (4 e^{8 t}+3 e^{12 t}-3\right )-6 c_2 \left (2 e^{8 t}+e^{12 t}-3\right )+4 c_3 e^{6 t}+32 c_3 e^{8 t}+12 c_3 e^{12 t}+4 c_4 e^{6 t}+44 c_4 e^{8 t}+15 c_4 e^{12 t}-48 c_3-63 c_4\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-4 t} \left (-\left (c_1 \left (-2 e^{8 t}+e^{12 t}+1\right )\right )+2 c_2 \left (-3 e^{8 t}+e^{12 t}+3\right )+4 c_3 e^{6 t}+16 c_3 e^{8 t}-4 c_3 e^{12 t}+4 c_4 e^{6 t}+22 c_4 e^{8 t}-5 c_4 e^{12 t}-16 c_3-21 c_4\right ) \\ \text {x3}(t)\to \frac {1}{4} e^{-4 t} \left (c_1 \left (-4 e^{8 t}+3 e^{12 t}+1\right )-6 c_2 \left (-2 e^{8 t}+e^{12 t}+1\right )+8 c_3 e^{6 t}-32 c_3 e^{8 t}+12 c_3 e^{12 t}+8 c_4 e^{6 t}-44 c_4 e^{8 t}+15 c_4 e^{12 t}+16 c_3+21 c_4\right ) \\ \text {x4}(t)\to \frac {1}{4} e^{-4 t} \left (c_1 \left (4 e^{8 t}-3 e^{12 t}-1\right )+6 c_2 \left (-2 e^{8 t}+e^{12 t}+1\right )-4 c_3 e^{6 t}+32 c_3 e^{8 t}-12 c_3 e^{12 t}-4 c_4 e^{6 t}+44 c_4 e^{8 t}-15 c_4 e^{12 t}-16 c_3-21 c_4\right ) \\ \end{align*}