Internal problem ID [357]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Examples. Page 437
Problem number: Example 6.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{3} \relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{4} \relax (t )\\ x_{3}^{\prime }\relax (t )&=-2 x_{1} \relax (t )+2 x_{2} \relax (t )-3 x_{3} \relax (t )+x_{4} \relax (t )\\ x_{4}^{\prime }\relax (t )&=2 x_{1} \relax (t )-2 x_{2} \relax (t )+x_{3} \relax (t )-3 x_{4} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 87
dsolve([diff(x__1(t),t)=0*x__1(t)+0*x__2(t)+1*x__3(t)+0*x__4(t),diff(x__2(t),t)=0*x__1(t)+0*x__2(t)+0*x__3(t)+1*x__4(t),diff(x__3(t),t)=-2*x__1(t)+2*x__2(t)-3*x__3(t)+1*x__4(t),diff(x__4(t),t)=2*x__1(t)-2*x__2(t)+1*x__3(t)-3*x__4(t)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1} \relax (t ) = \frac {c_{4} {\mathrm e}^{-2 t} t}{2}-\frac {{\mathrm e}^{-2 t} c_{2}}{2}+\frac {c_{3} {\mathrm e}^{-2 t}}{2}+\frac {c_{4} {\mathrm e}^{-2 t}}{4}+c_{1} \] \[ x_{2} \relax (t ) = \left (\left (-\frac {t}{2}-\frac {1}{4}\right ) c_{4}-\frac {c_{3}}{2}\right ) {\mathrm e}^{-2 t}+c_{1} \] \[ x_{3} \relax (t ) = {\mathrm e}^{-2 t} \left (-t c_{4}+c_{2}-c_{3}\right ) \] \[ x_{4} \relax (t ) = {\mathrm e}^{-2 t} \left (t c_{4}+c_{3}\right ) \]
✓ Solution by Mathematica
Time used: 0.059 (sec). Leaf size: 174
DSolve[{x1'[t]==0*x1[t]+0*x2[t]+1*x3[t]+0*x4[t],x2'[t]==0*x1[t]+0*x2[t]+0*x3[t]+1*x4[t],x3'[t]==-2*x1[t]+2*x2[t]-3*x3[t]+1*x4[t],x4'[t]==2*x1[t]-2*x2[t]+1*x3[t]-3*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to \frac {1}{4} \left (e^{-2 t} (c_1 (4 t+2)-2 c_2 (2 t+1)+c_3 (2 t-1)-c_4 (2 t+1))+2 c_1+2 c_2+c_3+c_4\right ) \\ \text {x2}(t)\to \frac {1}{4} \left (e^{-2 t} (c_4 (2 t-1)-(2 c_1-2 c_2+c_3) (2 t+1))+2 c_1+2 c_2+c_3+c_4\right ) \\ \text {x3}(t)\to e^{-2 t} ((-2 c_1+2 c_2-c_3+c_4) t+c_3) \\ \text {x4}(t)\to e^{-2 t} ((2 c_1-2 c_2+c_3-c_4) t+c_4) \\ \end{align*}