Internal problem ID [379]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number: problem 22.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1}\relax (t )+3 x_{2}\relax (t )+7 x_{3}\relax (t )\\ x_{2}^{\prime }\relax (t )&=-x_{2}\relax (t )-4 x_{3}\relax (t )\\ x_{3}^{\prime }\relax (t )&=x_{2}\relax (t )+3 x_{3}\relax (t )\\ x_{4}^{\prime }\relax (t )&=-6 x_{2}\relax (t )-14 x_{3}\relax (t )+x_{4}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.029 (sec). Leaf size: 73
dsolve([diff(x__1(t),t)=1*x__1(t)+3*x__2(t)+7*x__3(t)+0*x__4(t),diff(x__2(t),t)=0*x__1(t)-1*x__2(t)-4*x__3(t)+0*x__4(t),diff(x__3(t),t)=0*x__1(t)+1*x__2(t)+3*x__3(t)+0*x__4(t),diff(x__4(t),t)=0*x__1(t)-6*x__2(t)-14*x__3(t)+1*x__4(t)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1}\relax (t ) = \frac {{\mathrm e}^{t} \left (-c_{4} t^{2}-c_{3} t +2 c_{1}-c_{2}\right )}{2} \] \[ x_{2}\relax (t ) = {\mathrm e}^{t} \left (2 c_{4} t +c_{3}-7 c_{4}\right ) \] \[ x_{3}\relax (t ) = -\frac {{\mathrm e}^{t} \left (2 c_{4} t +c_{3}-6 c_{4}\right )}{2} \] \[ x_{4}\relax (t ) = {\mathrm e}^{t} \left (c_{4} t^{2}+c_{3} t +c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 97
DSolve[{x1'[t]==1*x1[t]+3*x2[t]+7*x3[t]+0*x4[t],x2'[t]==0*x1[t]-1*x2[t]-4*x3[t]+0*x4[t],x3'[t]==0*x1[t]+1*x2[t]+3*x3[t]+0*x4[t],x4'[t]==0*x1[t]-6*x2[t]-14*x3[t]+1*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{2} e^t (c_2 t (t+6)+2 c_3 t (t+7)+2 c_1) \\ \text {x2}(t)\to e^t (-2 c_2 t-4 c_3 t+c_2) \\ \text {x3}(t)\to e^t ((c_2+2 c_3) t+c_3) \\ \text {x4}(t)\to e^t (c_2 (-t) (t+6)-2 c_3 t (t+7)+c_4) \\ \end{align*}