18.3 problem 3

Internal problem ID [780]

Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section: Chapter 7.9, Nonhomogeneous Linear Systems. page 447
Problem number: 3.
ODE order: 1.
ODE degree: 1.

Solve \begin {align*} x_{1}^{\prime }\relax (t )&=2 x_{1} \relax (t )-5 x_{2} \relax (t )-\cos \relax (t )\\ x_{2}^{\prime }\relax (t )&=x_{1} \relax (t )-2 x_{2} \relax (t )+\sin \relax (t ) \end {align*}

Solution by Maple

Time used: 0.078 (sec). Leaf size: 59

dsolve([diff(x__1(t),t)=2*x__1(t)-5*x__2(t)-cos(t),diff(x__2(t),t)=1*x__1(t)-2*x__2(t)+sin(t)],[x__1(t), x__2(t)], singsol=all)
 

\[ x_{1} \relax (t ) = c_{2} \cos \relax (t )-\sin \relax (t ) c_{1}-\sin \relax (t ) t +2 \sin \relax (t ) c_{2}+2 \cos \relax (t ) c_{1}+2 \cos \relax (t ) t -3 \sin \relax (t ) \] \[ x_{2} \relax (t ) = \sin \relax (t ) c_{2}+\cos \relax (t ) c_{1}+\cos \relax (t ) t -\sin \relax (t ) \]

Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 61

DSolve[{x1'[t]==2*x1[t]-5*x2[t]-Cos[t],x2'[t]==1*x1[t]-2*x2[t]+Sin[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \left (2 t-\frac {1}{2}+c_1\right ) \cos (t)-(t-1-2 c_1+5 c_2) \sin (t) \\ \text {x2}(t)\to (t-1+c_2) \cos (t)+\frac {1}{2} (1+2 c_1-4 c_2) \sin (t) \\ \end{align*}