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 }\relax (t )&=13 x_{1} \relax (t )-42 x_{2} \relax (t )+106 x_{3} \relax (t )+139 x_{4} \relax (t )\\ x_{2}^{\prime }\relax (t )&=2 x_{1} \relax (t )-16 x_{2} \relax (t )+52 x_{3} \relax (t )+70 x_{4} \relax (t )\\ x_{3}^{\prime }\relax (t )&=x_{1} \relax (t )+6 x_{2} \relax (t )-20 x_{3} \relax (t )-31 x_{4} \relax (t )\\ x_{4}^{\prime }\relax (t )&=-x_{1} \relax (t )-6 x_{2} \relax (t )+22 x_{3} \relax (t )+33 x_{4} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 124
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)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1} \relax (t ) = c_{1} {\mathrm e}^{4 t}-c_{2} {\mathrm e}^{8 t}-c_{3} {\mathrm e}^{2 t}+3 c_{4} {\mathrm e}^{-4 t} \] \[ x_{2} \relax (t ) = c_{1} {\mathrm e}^{4 t}+\frac {2 c_{2} {\mathrm e}^{8 t}}{3}-2 c_{3} {\mathrm e}^{2 t}+2 c_{4} {\mathrm e}^{-4 t} \] \[ x_{3} \relax (t ) = -c_{1} {\mathrm e}^{4 t}-c_{2} {\mathrm e}^{8 t}-2 c_{3} {\mathrm e}^{2 t}-c_{4} {\mathrm e}^{-4 t} \] \[ x_{4} \relax (t ) = c_{1} {\mathrm e}^{4 t}+c_{2} {\mathrm e}^{8 t}+c_{3} {\mathrm e}^{2 t}+c_{4} {\mathrm e}^{-4 t} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 339
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 (c_3+c_4) e^{2 t}+\frac {3}{4} (c_1-2 c_2+4 c_3+5 c_4) e^{8 t}+(c_1-3 c_2+8 c_3+11 c_4) e^{4 t}-\frac {3}{4} (c_1-6 c_2+16 c_3+21 c_4) e^{-4 t} \\ \text {x2}(t)\to 2 (c_3+c_4) e^{2 t}-\frac {1}{2} (c_1-2 c_2+4 c_3+5 c_4) e^{8 t}+(c_1-3 c_2+8 c_3+11 c_4) e^{4 t}-\frac {1}{2} (c_1-6 c_2+16 c_3+21 c_4) e^{-4 t} \\ \text {x3}(t)\to \frac {1}{4} e^{-4 t} \left (8 (c_3+c_4) e^{6 t}+3 (c_1-2 c_2+4 c_3+5 c_4) e^{12 t}-4 (c_1-3 c_2+8 c_3+11 c_4) e^{8 t}+c_1-6 c_2+16 c_3+21 c_4\right ) \\ \text {x4}(t)\to (c_3+c_4) \left (-e^{2 t}\right )-\frac {3}{4} (c_1-2 c_2+4 c_3+5 c_4) e^{8 t}+(c_1-3 c_2+8 c_3+11 c_4) e^{4 t}-\frac {1}{4} (c_1-6 c_2+16 c_3+21 c_4) e^{-4 t} \\ \end{align*}