Internal problem ID [350]
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 47.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=23 x_{1}\relax (t )-18 x_{2}\relax (t )-16 x_{3}\relax (t )\\ x_{2}^{\prime }\relax (t )&=-8 x_{1}\relax (t )+6 x_{2}\relax (t )+7 x_{3}\relax (t )+9 x_{4}\relax (t )\\ x_{3}^{\prime }\relax (t )&=34 x_{1}\relax (t )-27 x_{2}\relax (t )-26 x_{3}\relax (t )-9 x_{4}\relax (t )\\ x_{4}^{\prime }\relax (t )&=-26 x_{1}\relax (t )+21 x_{2}\relax (t )+25 x_{3}\relax (t )+12 x_{4}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.029 (sec). Leaf size: 122
dsolve([diff(x__1(t),t)=23*x__1(t)-18*x__2(t)-16*x__3(t)+0*x__4(t),diff(x__2(t),t)=-8*x__1(t)+6*x__2(t)+7*x__3(t)+9*x__4(t),diff(x__3(t),t)=34*x__1(t)-27*x__2(t)-26*x__3(t)-9*x__4(t),diff(x__4(t),t)=-26*x__1(t)+21*x__2(t)+25*x__3(t)+12*x__4(t)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1}\relax (t ) = -2 c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{3 t}+2 c_{3} {\mathrm e}^{6 t}-c_{4} {\mathrm e}^{9 t} \] \[ x_{2}\relax (t ) = -2 c_{1} {\mathrm e}^{-3 t}+2 c_{2} {\mathrm e}^{3 t}+c_{3} {\mathrm e}^{6 t}+c_{4} {\mathrm e}^{9 t} \] \[ x_{3}\relax (t ) = -c_{1} {\mathrm e}^{-3 t}-c_{2} {\mathrm e}^{3 t}+c_{3} {\mathrm e}^{6 t}-2 c_{4} {\mathrm e}^{9 t} \] \[ x_{4}\relax (t ) = c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{3 t}+c_{3} {\mathrm e}^{6 t}+c_{4} {\mathrm e}^{9 t} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 369
DSolve[{x1'[t]==23*x1[t]-18*x2[t]-16*x3[t]+0*x4[t],x2'[t]==-8*x1[t]+6*x2[t]+7*x3[t]+9*x4[t],x3'[t]==34*x1[t]-27*x2[t]-26*x3[t]-9*x4[t],x4'[t]==-26*x1[t]+21*x2[t]+25*x3[t]+12*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to 3 c_1 e^{3 t}+\frac {8}{3} c_1 e^{9 t}+2 (-c_1+c_2+c_3) e^{-3 t}-\frac {2}{3} e^{6 t} ((6 c_2+8 c_3+3 c_4) \cosh (3 t)-c_3 \sinh (3 t)+4 c_1-3 c_2-5 c_3-3 c_4) \\ \text {x2}(t)\to \frac {1}{3} e^{-3 t} \left (6 (3 c_1-2 c_2-3 c_3-c_4) e^{6 t}+(-4 c_1+3 c_2+5 c_3+3 c_4) e^{9 t}+(-8 c_1+6 c_2+7 c_3+3 c_4) e^{12 t}+6 (-c_1+c_2+c_3)\right ) \\ \text {x3}(t)\to (-c_1+c_2+c_3) e^{-3 t}+\frac {2}{3} (8 c_1-6 c_2-7 c_3-3 c_4) e^{9 t}+\left (-\frac {4 c_1}{3}+c_2+\frac {5 c_3}{3}+c_4\right ) e^{6 t}+(-3 c_1+2 c_2+3 c_3+c_4) e^{3 t} \\ \text {x4}(t)\to \frac {1}{3} e^{-3 t} \left (c_1 \left (9 e^{6 t}-4 e^{9 t}-8 e^{12 t}+3\right )+e^{9 t} (-2 c_3 \cosh (3 t)+2 (6 c_2+8 c_3+3 c_4) \sinh (3 t)+3 c_2+5 c_3+3 c_4)-3 (c_2+c_3)\right ) \\ \end{align*}