10.18.3 problem 3

Internal problem ID [1430]
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
Date solved : Monday, January 27, 2025 at 04:57:12 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-5 x_{2} \left (t \right )-\cos \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-2 x_{2} \left (t \right )+\sin \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.081 (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)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= \cos \left (t \right ) c_1 +2 \cos \left (t \right ) t +\sin \left (t \right ) c_2 -\sin \left (t \right ) t -\cos \left (t \right ) \\ x_{2} \left (t \right ) &= \frac {2 \cos \left (t \right ) c_1}{5}-\frac {c_2 \cos \left (t \right )}{5}+\cos \left (t \right ) t +\frac {c_1 \sin \left (t \right )}{5}+\frac {2 \sin \left (t \right ) c_2}{5}-\cos \left (t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.061 (sec). Leaf size: 61

DSolve[{D[ x1[t],t]==2*x1[t]-5*x2[t]-Cos[t],D[ x2[t],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*}