16.9 problem 9

Internal problem ID [759]

Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section: Chapter 7.6, Complex Eigenvalues. page 417
Problem number: 9.
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 ) \end {align*}

With initial conditions \[ [x_{1} \relax (0) = 1, x_{2} \relax (0) = 1] \]

Solution by Maple

Time used: 0.047 (sec). Leaf size: 32

dsolve([diff(x__1(t),t) = x__1(t)-5*x__2(t), diff(x__2(t),t) = x__1(t)-3*x__2(t), x__1(0) = 1, x__2(0) = 1],[x__1(t), x__2(t)], singsol=all)
 

\[ x_{1} \relax (t ) = {\mathrm e}^{-t} \left (\cos \relax (t )-3 \sin \relax (t )\right ) \] \[ x_{2} \relax (t ) = {\mathrm e}^{-t} \left (-\sin \relax (t )+\cos \relax (t )\right ) \]

Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 34

DSolve[{x1'[t]==1*x1[t]-5*x2[t],x2'[t]==1*x1[t]-3*x2[t]},{x1[0]==1,x2[0]==1},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-t} (\cos (t)-3 \sin (t)) \\ \text {x2}(t)\to e^{-t} (\cos (t)-\sin (t)) \\ \end{align*}