Internal problem ID [351]
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 48.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=47 x_{1} \left (t \right )-8 x_{2} \left (t \right )+5 x_{3} \left (t \right )-5 x_{4} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-10 x_{1} \left (t \right )+32 x_{2} \left (t \right )+18 x_{3} \left (t \right )-2 x_{4} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=139 x_{1} \left (t \right )-40 x_{2} \left (t \right )-167 x_{3} \left (t \right )-121 x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=-232 x_{1} \left (t \right )+64 x_{2} \left (t \right )+360 x_{3} \left (t \right )+248 x_{4} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 125
dsolve([diff(x__1(t),t)=47*x__1(t)-8*x__2(t)+5*x__3(t)-5*x__4(t),diff(x__2(t),t)=-10*x__1(t)+32*x__2(t)+18*x__3(t)-2*x__4(t),diff(x__3(t),t)=139*x__1(t)-40*x__2(t)-167*x__3(t)-121*x__4(t),diff(x__4(t),t)=-232*x__1(t)+64*x__2(t)+360*x__3(t)+248*x__4(t)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{48 t}+c_{2} {\mathrm e}^{16 t}+c_{3} {\mathrm e}^{32 t}+c_{4} {\mathrm e}^{64 t} \\ x_{2} \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{48 t}}{3}+2 c_{2} {\mathrm e}^{16 t}+\frac {5 c_{3} {\mathrm e}^{32 t}}{2}+c_{4} {\mathrm e}^{64 t} \\ x_{3} \left (t \right ) &= \frac {c_{1} {\mathrm e}^{48 t}}{3}-c_{2} {\mathrm e}^{16 t}+\frac {c_{3} {\mathrm e}^{32 t}}{2}+2 c_{4} {\mathrm e}^{64 t} \\ x_{4} \left (t \right ) &= \frac {2 c_{1} {\mathrm e}^{48 t}}{3}+2 c_{2} {\mathrm e}^{16 t}-\frac {c_{3} {\mathrm e}^{32 t}}{2}-3 c_{4} {\mathrm e}^{64 t} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 448
DSolve[{x1'[t]==47*x1[t]-8*x2[t]+5*x3[t]-5*x4[t],x2'[t]==-10*x1[t]+32*x2[t]+18*x3[t]-2*x4[t],x3'[t]==139*x1[t]-40*x2[t]-167*x3[t]-121*x4[t],x4'[t]==-232*x1[t]+64*x2[t]+360*x3[t]+248*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{16} e^{16 t} \left (c_1 \left (-38 e^{16 t}-6 e^{32 t}+27 e^{48 t}+33\right )-\left (e^{16 t}-1\right ) \left (8 c_2 \left (e^{16 t}+e^{32 t}-1\right )+c_3 \left (9 e^{16 t}+39 e^{32 t}-53\right )+c_4 \left (7 e^{16 t}+25 e^{32 t}-27\right )\right )\right ) \\ \text {x2}(t)\to \frac {1}{16} e^{16 t} \left (c_1 \left (-95 e^{16 t}+2 e^{32 t}+27 e^{48 t}+66\right )-8 c_2 \left (-5 e^{16 t}+e^{48 t}+2\right )-\left (e^{16 t}-1\right ) \left (c_3 \left (49 e^{16 t}+39 e^{32 t}-106\right )+c_4 \left (31 e^{16 t}+25 e^{32 t}-54\right )\right )\right ) \\ \text {x3}(t)\to \frac {1}{16} e^{16 t} \left (c_1 \left (-19 e^{16 t}-2 e^{32 t}+54 e^{48 t}-33\right )+8 c_2 \left (e^{16 t}-2 e^{48 t}+1\right )+31 c_3 e^{16 t}+10 c_3 e^{32 t}-78 c_3 e^{48 t}+17 c_4 e^{16 t}+6 c_4 e^{32 t}-50 c_4 e^{48 t}+53 c_3+27 c_4\right ) \\ \text {x4}(t)\to -\frac {1}{16} e^{16 t} \left (c_1 \left (-19 e^{16 t}+4 e^{32 t}+81 e^{48 t}-66\right )+8 c_2 \left (e^{16 t}-3 e^{48 t}+2\right )+31 c_3 e^{16 t}-20 c_3 e^{32 t}-117 c_3 e^{48 t}+17 c_4 e^{16 t}-12 c_4 e^{32 t}-75 c_4 e^{48 t}+106 c_3+54 c_4\right ) \\ \end{align*}