Internal problem ID [388]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number: problem 31.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=35 x_{1} \left (t \right )-12 x_{2} \left (t \right )+4 x_{3} \left (t \right )+30 x_{4} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=22 x_{1} \left (t \right )-8 x_{2} \left (t \right )+3 x_{3} \left (t \right )+19 x_{4} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=-10 x_{1} \left (t \right )+3 x_{2} \left (t \right )-9 x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=-27 x_{1} \left (t \right )+9 x_{2} \left (t \right )-3 x_{3} \left (t \right )-23 x_{4} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 117
dsolve([diff(x__1(t),t)=35*x__1(t)-12*x__2(t)+4*x__3(t)+30*x__4(t),diff(x__2(t),t)=22*x__1(t)-8*x__2(t)+3*x__3(t)+19*x__4(t),diff(x__3(t),t)=-10*x__1(t)+3*x__2(t)+0*x__3(t)-9*x__4(t),diff(x__4(t),t)=-27*x__1(t)+9*x__2(t)-3*x__3(t)-23*x__4(t)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-6 c_{4} t^{2}-6 c_{3} t -4 c_{4} t +3 c_{1} -6 c_{2} -2 c_{3} \right )}{3} \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{t} \left (-3 c_{4} t^{2}-3 c_{3} t -16 c_{4} t +3 c_{1} -3 c_{2} -8 c_{3} +6 c_{4} \right )}{9} \\ x_{3} \left (t \right ) &= {\mathrm e}^{t} \left (c_{4} t^{2}+c_{3} t +c_{2} \right ) \\ x_{4} \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (-18 c_{4} t^{2}-18 c_{3} t -6 c_{4} t +9 c_{1} -18 c_{2} -3 c_{3} -2 c_{4} \right )}{9} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 207
DSolve[{x1'[t]==35*x1[t]-12*x2[t]+4*x3[t]+30*x4[t],x2'[t]==22*x1[t]-8*x2[t]+3*x3[t]+19*x4[t],x3'[t]==-10*x1[t]+3*x2[t]+0*x3[t]-9*x4[t],x4'[t]==-27*x1[t]+9*x2[t]-3*x3[t]-23*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^t \left (c_1 \left (21 t^2+34 t+1\right )-3 c_2 t (3 t+4)+c_3 t (3 t+4)+6 c_4 t (3 t+5)\right ) \\ \text {x2}(t)\to \frac {1}{2} e^t \left ((7 c_1-3 c_2+c_3+6 c_4) t^2+2 (22 c_1-9 c_2+3 c_3+19 c_4) t+2 c_2\right ) \\ \text {x3}(t)\to \frac {1}{2} e^t \left (-3 (7 c_1-3 c_2+c_3+6 c_4) t^2-2 (10 c_1-3 c_2+c_3+9 c_4) t+2 c_3\right ) \\ \text {x4}(t)\to e^t \left (-3 (7 c_1-3 c_2+c_3+6 c_4) t^2-3 (9 c_1-3 c_2+c_3+8 c_4) t+c_4\right ) \\ \end{align*}