Internal problem ID [391]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number: problem 34.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-8 x_{3} \left (t \right )-3 x_{4} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-18 x_{1} \left (t \right )-x_{2} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=-9 x_{1} \left (t \right )-3 x_{2} \left (t \right )-25 x_{3} \left (t \right )-9 x_{4} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=33 x_{1} \left (t \right )+10 x_{2} \left (t \right )+90 x_{3} \left (t \right )+32 x_{4} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 252
dsolve([diff(x__1(t),t)=2*x__1(t)+0*x__2(t)-8*x__3(t)-3*x__4(t),diff(x__2(t),t)=-18*x__1(t)-1*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__3(t),t)=-9*x__1(t)-3*x__2(t)-25*x__3(t)-9*x__4(t),diff(x__4(t),t)=33*x__1(t)+10*x__2(t)+90*x__3(t)+32*x__4(t)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (3 \cos \left (3 t \right ) c_{3} t +3 \cos \left (3 t \right ) c_{4} t +3 \sin \left (3 t \right ) c_{3} t -3 \sin \left (3 t \right ) c_{4} t +3 c_{1} \cos \left (3 t \right )+3 c_{2} \cos \left (3 t \right )+\cos \left (3 t \right ) c_{4} +3 c_{1} \sin \left (3 t \right )-3 c_{2} \sin \left (3 t \right )+\sin \left (3 t \right ) c_{3} \right )}{18} \\ x_{2} \left (t \right ) &= {\mathrm e}^{2 t} \left (\cos \left (3 t \right ) c_{4} t +\sin \left (3 t \right ) c_{3} t +c_{2} \cos \left (3 t \right )+c_{1} \sin \left (3 t \right )\right ) \\ x_{3} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (\cos \left (3 t \right ) c_{3} +\cos \left (3 t \right ) c_{4} +\sin \left (3 t \right ) c_{3} -\sin \left (3 t \right ) c_{4} \right )}{6} \\ x_{4} \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (3 \cos \left (3 t \right ) c_{3} t -3 \cos \left (3 t \right ) c_{4} t -3 \sin \left (3 t \right ) c_{3} t -3 \sin \left (3 t \right ) c_{4} t +3 c_{1} \cos \left (3 t \right )-3 c_{2} \cos \left (3 t \right )+10 \cos \left (3 t \right ) c_{3} +9 \cos \left (3 t \right ) c_{4} -3 c_{1} \sin \left (3 t \right )-3 c_{2} \sin \left (3 t \right )+9 \sin \left (3 t \right ) c_{3} -10 \sin \left (3 t \right ) c_{4} \right )}{18} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.026 (sec). Leaf size: 482
DSolve[{x1'[t]==2*x1[t]+0*x2[t]-8*x3[t]-3*x4[t],x2'[t]==-18*x1[t]-1*x2[t]+0*x3[t]+0*x4[t],x3'[t]==-9*x1[t]-3*x2[t]-25*x3[t]-9*x4[t],x4'[t]==33*x1[t]+10*x2[t]+90*x3[t]+32*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{(2-3 i) t} \left (c_1 \left (e^{6 i t} (1+3 i t)-3 i t+1\right )+i (3 c_3+c_4) \left (-1+e^{6 i t}\right )+t \left (i c_2 \left (-1+e^{6 i t}\right )+c_3 \left ((1+9 i) e^{6 i t}+(1-9 i)\right )+3 i c_4 \left (-1+e^{6 i t}\right )\right )\right ) \\ \text {x2}(t)\to -\frac {1}{2} e^{(2-3 i) t} \left (c_1 \left ((9-9 i) t+e^{6 i t} ((9+9 i) t-3 i)+3 i\right )+c_2 \left ((3-3 i) t+e^{6 i t} (-1+(3+3 i) t)-1\right )+10 i c_3 e^{6 i t}+(30+24 i) c_3 e^{6 i t} t+(30-24 i) c_3 t+3 i c_4 e^{6 i t}+(9+9 i) c_4 e^{6 i t} t+(9-9 i) c_4 t-10 i c_3-3 i c_4\right ) \\ \text {x3}(t)\to \frac {1}{2} e^{(2-3 i) t} \left (3 i c_1 \left (-1+e^{6 i t}\right )+i c_2 \left (-1+e^{6 i t}\right )+(1+9 i) c_3 e^{6 i t}+3 i c_4 e^{6 i t}+(1-9 i) c_3-3 i c_4\right ) \\ \text {x4}(t)\to \frac {1}{2} e^{(2-3 i) t} \left (c_1 \left (3 t+e^{6 i t} (3 t-10 i)+10 i\right )+c_2 \left (t+e^{6 i t} (t-3 i)+3 i\right )-27 i c_3 e^{6 i t}+(9-i) c_3 e^{6 i t} t+(9+i) c_3 t+(1-9 i) c_4 e^{6 i t}+3 c_4 e^{6 i t} t+3 c_4 t+27 i c_3+(1+9 i) c_4\right ) \\ \end{align*}