Internal problem ID [342]
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 39.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=-2 x_{1}\relax (t )+9 x_{4}\relax (t )\\ x_{2}^{\prime }\relax (t )&=4 x_{1}\relax (t )+2 x_{2}\relax (t )-10 x_{4}\relax (t )\\ x_{3}^{\prime }\relax (t )&=-x_{3}\relax (t )+8 x_{4}\relax (t )\\ x_{4}^{\prime }\relax (t )&=x_{4}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.057 (sec). Leaf size: 61
dsolve([diff(x__1(t),t)=-2*x__1(t)+0*x__2(t)+0*x__3(t)+9*x__4(t),diff(x__2(t),t)=4*x__1(t)+2*x__2(t)+0*x__3(t)-10*x__4(t),diff(x__3(t),t)=0*x__1(t)+0*x__2(t)-1*x__3(t)+8*x__4(t),diff(x__4(t),t)=0*x__1(t)+0*x__2(t)+0*x__3(t)+1*x__4(t)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\[ x_{1}\relax (t ) = -{\mathrm e}^{-2 t} c_{2}+3 c_{4} {\mathrm e}^{t} \] \[ x_{2}\relax (t ) = {\mathrm e}^{-2 t} c_{2}+c_{1} {\mathrm e}^{2 t}-2 c_{4} {\mathrm e}^{t} \] \[ x_{3}\relax (t ) = 4 c_{4} {\mathrm e}^{t}+c_{3} {\mathrm e}^{-t} \] \[ x_{4}\relax (t ) = c_{4} {\mathrm e}^{t} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 94
DSolve[{x1'[t]==-2*x1[t]+0*x2[t]+0*x3[t]+9*x4[t],x2'[t]==4*x1[t]+2*x2[t]+0*x3[t]-10*x4[t],x3'[t]==0*x1[t]+0*x2[t]-1*x3[t]+8*x4[t],x4'[t]==0*x1[t]+0*x2[t]+0*x3[t]+1*x4[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^{-2 t} \left (3 c_4 \left (e^{3 t}-1\right )+c_1\right ) \\ \text {x2}(t)\to (c_1-3 c_4) \left (-e^{-2 t}\right )+(c_1+c_2-c_4) e^{2 t}-2 c_4 e^t \\ \text {x3}(t)\to c_3 \cosh (t)-(c_3-8 c_4) \sinh (t) \\ \text {x4}(t)\to c_4 e^t \\ \end{align*}