Internal problem ID [387]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number: problem 30.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=2 x_{1}\relax (t )+x_{2}\relax (t )-2 x_{3}\relax (t )+x_{4}\relax (t )\\ x_{2}^{\prime }\relax (t )&=3 x_{2}\relax (t )-5 x_{3}\relax (t )+3 x_{4}\relax (t )\\ x_{3}^{\prime }\relax (t )&=-13 x_{2}\relax (t )+22 x_{3}\relax (t )-12 x_{4}\relax (t )\\ x_{4}^{\prime }\relax (t )&=-27 x_{2}\relax (t )+45 x_{3}\relax (t )-25 x_{4}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.03 (sec). Leaf size: 95
dsolve([diff(x__1(t),t)=2*x__1(t)+1*x__2(t)-2*x__3(t)+1*x__4(t),diff(x__2(t),t)=0*x__1(t)+3*x__2(t)-5*x__3(t)+3*x__4(t),diff(x__3(t),t)=0*x__1(t)-13*x__2(t)+22*x__3(t)-12*x__4(t),diff(x__4(t),t)=0*x__1(t)-27*x__2(t)+45*x__3(t)-25*x__4(t)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1}\relax (t ) = \frac {\left (-c_{2} t +5 c_{1}\right ) {\mathrm e}^{2 t}}{5} \] \[ x_{2}\relax (t ) = -\frac {{\mathrm e}^{-t} \left (3 c_{4} t +3 c_{3}+2 c_{4}\right )}{9} \] \[ x_{3}\relax (t ) = \frac {3 c_{2} {\mathrm e}^{2 t}}{5}+\frac {c_{3} {\mathrm e}^{-t}}{3}+\frac {c_{4} {\mathrm e}^{-t} t}{3}-\frac {{\mathrm e}^{-t} c_{4}}{9} \] \[ x_{4}\relax (t ) = c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{-t}+c_{4} {\mathrm e}^{-t} t \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 157
DSolve[{x1'[t]==2*x1[t]+1*x2[t]-2*x3[t]+1*x4[t],x2'[t]==0*x1[t]+3*x2[t]-5*x3[t]+3*x4[t],x3'[t]==0*x1[t]-13*x2[t]+22*x3[t]-12*x4[t],x4'[t]==0*x1[t]-27*x2[t]+45*x3[t]-25*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^{2 t} ((c_2-2 c_3+c_4) t+c_1) \\ \text {x2}(t)\to e^{-t} (4 c_2 t-5 c_3 t+3 c_4 t+c_2) \\ \text {x3}(t)\to e^{-t} \left (c_2 \left (-4 t-3 e^{3 t}+3\right )+c_3 \left (5 t+6 e^{3 t}-5\right )-3 c_4 \left (t+e^{3 t}-1\right )\right ) \\ \text {x4}(t)\to e^{-t} \left (-12 c_2 t+15 c_3 t-9 c_4 t-5 (c_2-2 c_3+c_4) e^{3 t}+5 c_2-10 c_3+6 c_4\right ) \\ \end{align*}