Internal problem ID [348]
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 45.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=9 x_{1} \left (t \right )-7 x_{2} \left (t \right )-5 x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-12 x_{1} \left (t \right )+7 x_{2} \left (t \right )+11 x_{3} \left (t \right )+9 x_{4} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=24 x_{1} \left (t \right )-17 x_{2} \left (t \right )-19 x_{3} \left (t \right )-9 x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=-18 x_{1} \left (t \right )+13 x_{2} \left (t \right )+17 x_{3} \left (t \right )+9 x_{4} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 105
dsolve([diff(x__1(t),t)=9*x__1(t)-7*x__2(t)-5*x__3(t)+0*x__4(t),diff(x__2(t),t)=-12*x__1(t)+7*x__2(t)+11*x__3(t)+9*x__4(t),diff(x__3(t),t)=24*x__1(t)-17*x__2(t)-19*x__3(t)-9*x__4(t),diff(x__4(t),t)=-18*x__1(t)+13*x__2(t)+17*x__3(t)+9*x__4(t)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{3 t}+c_{3} {\mathrm e}^{6 t}+c_{4} {\mathrm e}^{-3 t} \\ x_{2} \left (t \right ) &= \frac {c_{2} {\mathrm e}^{3 t}}{2}-c_{3} {\mathrm e}^{6 t}+c_{4} {\mathrm e}^{-3 t}+2 c_{1} \\ x_{3} \left (t \right ) &= \frac {c_{2} {\mathrm e}^{3 t}}{2}+2 c_{3} {\mathrm e}^{6 t}+c_{4} {\mathrm e}^{-3 t}-c_{1} \\ x_{4} \left (t \right ) &= \frac {c_{2} {\mathrm e}^{3 t}}{2}-c_{3} {\mathrm e}^{6 t}-c_{4} {\mathrm e}^{-3 t}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 430
DSolve[{x1'[t]==9*x1[t]-7*x2[t]-5*x3[t]+0*x4[t],x2'[t]==-12*x1[t]+7*x2[t]+11*x3[t]+9*x4[t],x3'[t]==24*x1[t]-17*x2[t]-19*x3[t]-9*x4[t],x4'[t]==-18*x1[t]+13*x2[t]+17*x3[t]+9*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-3 t} \left (c_1 \left (6 e^{3 t}-6 e^{6 t}+6 e^{9 t}-3\right )-\left (e^{3 t}-1\right ) \left (c_2 \left (4 e^{6 t}+3\right )+c_3 \left (-3 e^{3 t}+5 e^{6 t}+3\right )+3 c_4 e^{3 t} \left (e^{3 t}-1\right )\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-3 t} \left (-3 c_1 \left (-4 e^{3 t}+e^{6 t}+2 e^{9 t}+1\right )+c_2 \left (-6 e^{3 t}+2 e^{6 t}+4 e^{9 t}+3\right )+\left (e^{3 t}-1\right ) \left (c_3 \left (9 e^{3 t}+5 e^{6 t}-3\right )+3 c_4 e^{3 t} \left (e^{3 t}+2\right )\right )\right ) \\ \text {x3}(t)\to c_1 \left (-e^{-3 t}-e^{3 t}+4 e^{6 t}-2\right )+c_2 \left (e^{-3 t}+\frac {2 e^{3 t}}{3}-\frac {8 e^{6 t}}{3}+1\right )+c_3 e^{-3 t}+\frac {4}{3} c_3 e^{3 t}-\frac {10}{3} c_3 e^{6 t}+c_4 e^{3 t}-2 c_4 e^{6 t}+2 c_3+c_4 \\ \text {x4}(t)\to \frac {1}{3} \left (c_1 \left (3 e^{-3 t}-3 e^{3 t}-6 e^{6 t}+6\right )+c_2 \left (-3 e^{-3 t}+2 e^{3 t}+4 e^{6 t}-3\right )-3 c_3 e^{-3 t}+4 c_3 e^{3 t}+5 c_3 e^{6 t}+3 c_4 e^{3 t}+3 c_4 e^{6 t}-6 c_3-3 c_4\right ) \\ \end{align*}