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 }\relax (t )&=2 x_{1}\relax (t )-8 x_{3}\relax (t )-3 x_{4}\relax (t )\\ x_{2}^{\prime }\relax (t )&=-18 x_{1}\relax (t )-x_{2}\relax (t )\\ x_{3}^{\prime }\relax (t )&=-9 x_{1}\relax (t )-3 x_{2}\relax (t )-25 x_{3}\relax (t )-9 x_{4}\relax (t )\\ x_{4}^{\prime }\relax (t )&=33 x_{1}\relax (t )+10 x_{2}\relax (t )+90 x_{3}\relax (t )+32 x_{4}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.042 (sec). Leaf size: 208
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)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1}\relax (t ) = {\mathrm e}^{2 t} \left (\cos \left (3 t \right ) c_{3} t -\sin \left (3 t \right ) c_{4} t +c_{1} \cos \left (3 t \right )-3 \cos \left (3 t \right ) c_{4}-c_{2} \sin \left (3 t \right )-3 \sin \left (3 t \right ) c_{3}\right ) \] \[ x_{2}\relax (t ) = -{\mathrm e}^{2 t} \left (3 \cos \left (3 t \right ) c_{3} t +3 \cos \left (3 t \right ) c_{4} t +3 \sin \left (3 t \right ) c_{3} t -3 \sin \left (3 t \right ) c_{4} t +3 c_{1} \cos \left (3 t \right )+3 c_{2} \cos \left (3 t \right )+9 c_{3} \cos \left (3 t \right )-10 \cos \left (3 t \right ) c_{4}+3 c_{1} \sin \left (3 t \right )-3 c_{2} \sin \left (3 t \right )-10 \sin \left (3 t \right ) c_{3}-9 \sin \left (3 t \right ) c_{4}\right ) \] \[ x_{3}\relax (t ) = {\mathrm e}^{2 t} \left (c_{3} \cos \left (3 t \right )-\sin \left (3 t \right ) c_{4}\right ) \] \[ x_{4}\relax (t ) = {\mathrm e}^{2 t} \left (\cos \left (3 t \right ) c_{4} t +\sin \left (3 t \right ) c_{3} t +c_{2} \cos \left (3 t \right )+c_{1} \sin \left (3 t \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.027 (sec). Leaf size: 223
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 e^{2 t} ((c_3 t+c_1) \cos (3 t)-((3 c_1+c_2+9 c_3+3 c_4) t+3 c_3+c_4) \sin (3 t)) \\ \text {x2}(t)\to e^{2 t} ((c_2-3 (3 c_1+c_2+10 c_3+3 c_4) t) \cos (3 t)+(c_1 (9 t-3)+3 (c_2+8 c_3+3 c_4) t+10 c_3+3 c_4) \sin (3 t)) \\ \text {x3}(t)\to e^{2 t} (c_3 \cos (3 t)-(3 c_1+c_2+9 c_3+3 c_4) \sin (3 t)) \\ \text {x4}(t)\to e^{2 t} (((3 c_1+c_2+9 c_3+3 c_4) t+c_4) \cos (3 t)+(c_3 (t+27)+10 c_1+3 c_2+9 c_4) \sin (3 t)) \\ \end{align*}