76.28.3 problem 4

Internal problem ID [17870]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.6 (Nonhomogeneous Linear Systems). Problems at page 436
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 11:04:18 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: 0.374 (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 ) &= c_{2} \sin \left (t \right )-t \sin \left (t \right )+\cos \left (t \right ) c_{1} +2 t \cos \left (t \right )-\cos \left (t \right ) \\ x_{2} \left (t \right ) &= \frac {c_{1} \sin \left (t \right )}{5}+\frac {2 c_{2} \sin \left (t \right )}{5}+\frac {2 \cos \left (t \right ) c_{1}}{5}-\frac {c_{2} \cos \left (t \right )}{5}+t \cos \left (t \right )-\cos \left (t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.042 (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*}