Internal problem ID [343]
Book: Differential equations and linear algebra, 4th ed., Edwards and Penney
Section: Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number: problem 40.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=2 x_{1}\relax (t )\\ x_{2}^{\prime }\relax (t )&=-21 x_{1}\relax (t )-5 x_{2}\relax (t )-27 x_{3}\relax (t )-9 x_{4}\relax (t )\\ x_{3}^{\prime }\relax (t )&=5 x_{3}\relax (t )\\ x_{4}^{\prime }\relax (t )&=-21 x_{3}\relax (t )-2 x_{4}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.058 (sec). Leaf size: 61
dsolve([diff(x__1(t),t)=2*x__1(t)+0*x__2(t)+0*x__3(t)+0*x__4(t),diff(x__2(t),t)=-21*x__1(t)-5*x__2(t)-27*x__3(t)-9*x__4(t),diff(x__3(t),t)=0*x__1(t)+0*x__2(t)+5*x__3(t)+0*x__4(t),diff(x__4(t),t)=0*x__1(t)+0*x__2(t)-21*x__3(t)-2*x__4(t)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1}\relax (t ) = -\frac {c_{1} {\mathrm e}^{2 t}}{3} \] \[ x_{2}\relax (t ) = {\mathrm e}^{-5 t} c_{2}+c_{1} {\mathrm e}^{2 t}-3 c_{3} {\mathrm e}^{-2 t} \] \[ x_{3}\relax (t ) = -\frac {c_{4} {\mathrm e}^{5 t}}{3} \] \[ x_{4}\relax (t ) = c_{3} {\mathrm e}^{-2 t}+c_{4} {\mathrm e}^{5 t} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 86
DSolve[{x1'[t]==2*x1[t]+0*x2[t]+0*x3[t]+0*x4[t],x2'[t]==-21*x1[t]-5*x2[t]-27*x3[t]-9*x4[t],x3'[t]==0*x1[t]+0*x2[t]+5*x3[t]+0*x4[t],x4'[t]==0*x1[t]+0*x2[t]-21*x3[t]-2*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to c_1 e^{2 t} \\ \text {x2}(t)\to e^{-5 t} \left (-3 c_1 \left (e^{7 t}-1\right )-3 (3 c_3+c_4) \left (e^{3 t}-1\right )+c_2\right ) \\ \text {x3}(t)\to c_3 e^{5 t} \\ \text {x4}(t)\to e^{-2 t} \left (c_4-3 c_3 \left (e^{7 t}-1\right )\right ) \\ \end{align*}