Internal problem ID [389]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number: problem 32.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=11 x_{1} \relax (t )-x_{2} \relax (t )+26 x_{3} \relax (t )+6 x_{4} \relax (t )-3 x_{5} \relax (t )\\ x_{2}^{\prime }\relax (t )&=3 x_{2} \relax (t )\\ x_{3}^{\prime }\relax (t )&=-9 x_{1} \relax (t )-24 x_{3} \relax (t )-6 x_{4} \relax (t )+3 x_{5} \relax (t )\\ x_{4}^{\prime }\relax (t )&=3 x_{1} \relax (t )+9 x_{3} \relax (t )+5 x_{4} \relax (t )-x_{5} \relax (t )\\ x_{5}^{\prime }\relax (t )&=-48 x_{1} \relax (t )-3 x_{2} \relax (t )-138 x_{3} \relax (t )-30 x_{4} \relax (t )+18 x_{5} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 116
dsolve([diff(x__1(t),t)=11*x__1(t)-1*x__2(t)+26*x__3(t)+6*x__4(t)-3*x__5(t),diff(x__2(t),t)=0*x__1(t)+3*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__3(t),t)=-9*x__1(t)+0*x__2(t)-24*x__3(t)-6*x__4(t)+3*x__5(t),diff(x__4(t),t)=3*x__1(t)+0*x__2(t)+9*x__3(t)+5*x__4(t)-1*x__5(t),diff(x__5(t),t)=-48*x__1(t)-3*x__2(t)-138*x__3(t)-30*x__4(t)+18*x__5(t)],[x__1(t), x__2(t), x__3(t), x__4(t), x__5(t)], singsol=all)
\[ x_{1} \relax (t ) = 8 c_{2} {\mathrm e}^{2 t}+\frac {25 c_{3} {\mathrm e}^{3 t}}{3}-3 c_{1} {\mathrm e}^{3 t}+\frac {c_{4} {\mathrm e}^{2 t}}{3}+\frac {c_{5} {\mathrm e}^{3 t}}{3} \] \[ x_{2} \relax (t ) = \frac {{\mathrm e}^{3 t} \left (6 c_{1}-16 c_{3}-c_{5}\right )}{3} \] \[ x_{3} \relax (t ) = -3 c_{2} {\mathrm e}^{2 t}-3 c_{3} {\mathrm e}^{3 t}+c_{1} {\mathrm e}^{3 t} \] \[ x_{4} \relax (t ) = c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{3 t} \] \[ x_{5} \relax (t ) = c_{4} {\mathrm e}^{2 t}+c_{5} {\mathrm e}^{3 t} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 202
DSolve[{x1'[t]==11*x1[t]-1*x2[t]+26*x3[t]+6*x4[t]-3*x5[t],x2'[t]==0*x1[t]+3*x2[t],x3'[t]==-9*x1[t]+0*x2[t]-24*x3[t]-6*x4[t]+3*x5[t],x4'[t]==3*x1[t]+0*x2[t]+9*x3[t]+5*x4[t]-1*x5[t],x5'[t]==-48*x1[t]-3*x2[t]-138*x3[t]-30*x4[t]+18*x5[t]},{x1[t],x2[t],x3[t],x4[t],x5[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^{2 t} \left (c_1 \left (9 e^t-8\right )-(c_2-26 c_3-6 c_4+3 c_5) \left (e^t-1\right )\right ) \\ \text {x2}(t)\to c_2 e^{3 t} \\ \text {x3}(t)\to -e^{2 t} \left (9 c_1 \left (e^t-1\right )+c_3 \left (26 e^t-27\right )+3 (2 c_4-c_5) \left (e^t-1\right )\right ) \\ \text {x4}(t)\to e^{2 t} \left ((3 (c_1+3 c_3+c_4)-c_5) e^t-3 c_1-9 c_3-2 c_4+c_5\right ) \\ \text {x5}(t)\to e^{2 t} \left (3 (16 c_1+c_2+46 c_3+10 c_4-5 c_5)-(48 c_1+3 c_2+138 c_3+30 c_4-16 c_5) e^t\right ) \\ \end{align*}