76.6.17 problem 19

Internal problem ID [17472]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 19
Date solved : Tuesday, January 28, 2025 at 10:39:13 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+y \left (t \right )-3\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+y \left (t \right )+1 \end{align*}

✓ Solution by Maple

Time used: 0.074 (sec). Leaf size: 37

dsolve([diff(x(t),t)=x(t)+y(t)-3,diff(y(t),t)=-x(t)+y(t)+1],singsol=all)
 

\begin{align*} x \left (t \right ) &= 2+{\mathrm e}^{t} \left (c_{2} \sin \left (t \right )+\cos \left (t \right ) c_{1} \right ) \\ y \left (t \right ) &= 1-{\mathrm e}^{t} \left (-c_{2} \cos \left (t \right )+c_{1} \sin \left (t \right )\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 146

DSolve[{D[x[t],t]==x[t]+y[t]-3,D[y[t],t]==-x[t]+y[t]+1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^t \left (\cos (t) \int _1^t-e^{-K[1]} (3 \cos (K[1])+\sin (K[1]))dK[1]+\sin (t) \int _1^te^{-K[2]} (\cos (K[2])-3 \sin (K[2]))dK[2]+c_1 \cos (t)+c_2 \sin (t)\right ) \\ y(t)\to e^t \left (\cos (t) \int _1^te^{-K[2]} (\cos (K[2])-3 \sin (K[2]))dK[2]-\sin (t) \int _1^t-e^{-K[1]} (3 \cos (K[1])+\sin (K[1]))dK[1]+c_2 \cos (t)-c_1 \sin (t)\right ) \\ \end{align*}