Internal problem ID [1837]
Book: Differential equations and their applications, 4th ed., M. Braun
Section: Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number: 2.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\relax (t )&=x_{1}\relax (t )-5 x_{2}\relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{1}\relax (t )-3 x_{2}\relax (t )\\ x_{3}^{\prime }\relax (t )&=x_{3}\relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.049 (sec). Leaf size: 54
dsolve([diff(x__1(t),t)=1*x__1(t)-5*x__2(t)+0*x__3(t),diff(x__2(t),t)=1*x__1(t)-3*x__2(t)+0*x__3(t),diff(x__3(t),t)=0*x__1(t)-0*x__2(t)+1*x__3(t)],[x__1(t), x__2(t), x__3(t)], singsol=all)
\[ x_{1}\relax (t ) = {\mathrm e}^{-t} \left (\cos \relax (t ) c_{1}-\sin \relax (t ) c_{2}+2 c_{1} \sin \relax (t )+2 c_{2} \cos \relax (t )\right ) \] \[ x_{2}\relax (t ) = {\mathrm e}^{-t} \left (c_{1} \sin \relax (t )+c_{2} \cos \relax (t )\right ) \] \[ x_{3}\relax (t ) = c_{3} {\mathrm e}^{t} \]
✓ Solution by Mathematica
Time used: 0.025 (sec). Leaf size: 120
DSolve[{x1'[t]==1*x1[t]-5*x2[t]+0*x3[t],x2'[t]==1*x1[t]-3*x2[t]+0*x3[t],x3'[t]==0*x1[t]-0*x2[t]+1*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to e^{-t} (c_1 \cos (t)+(2 c_1-5 c_2) \sin (t)) \\ \text {x2}(t)\to e^{-t} (c_2 \cos (t)+(c_1-2 c_2) \sin (t)) \\ \text {x3}(t)\to c_3 e^t \\ \text {x1}(t)\to e^{-t} (c_1 \cos (t)+(2 c_1-5 c_2) \sin (t)) \\ \text {x2}(t)\to e^{-t} (c_2 \cos (t)+(c_1-2 c_2) \sin (t)) \\ \text {x3}(t)\to 0 \\ \end{align*}