76.6.14 problem 16

Internal problem ID [17469]
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 : 16
Date solved : Tuesday, January 28, 2025 at 10:39:10 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )-4 y \left (t \right )-4\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-y \left (t \right )-6 \end{align*}

✓ Solution by Maple

Time used: 0.072 (sec). Leaf size: 49

dsolve([diff(x(t),t)=-x(t)-4*y(t)-4,diff(y(t),t)=x(t)-y(t)-6],singsol=all)
 

\begin{align*} x \left (t \right ) &= 4+{\mathrm e}^{-t} \left (\sin \left (2 t \right ) c_{2} +\cos \left (2 t \right ) c_{1} \right ) \\ y \left (t \right ) &= -2-\frac {{\mathrm e}^{-t} \left (c_{2} \cos \left (2 t \right )-c_{1} \sin \left (2 t \right )\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.177 (sec). Leaf size: 185

DSolve[{D[x[t],t]==-x[t]-4*y[t]-4,D[y[t],t]==x[t]-y[t]-6},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-t} \left (\cos (2 t) \int _1^t-4 e^{K[1]} (\cos (2 K[1])+3 \sin (2 K[1]))dK[1]-2 \sin (2 t) \int _1^t-2 e^{K[2]} (3 \cos (2 K[2])-\sin (2 K[2]))dK[2]+c_1 \cos (2 t)-2 c_2 \sin (2 t)\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (2 \cos (2 t) \int _1^t-2 e^{K[2]} (3 \cos (2 K[2])-\sin (2 K[2]))dK[2]+\sin (2 t) \int _1^t-4 e^{K[1]} (\cos (2 K[1])+3 \sin (2 K[1]))dK[1]+2 c_2 \cos (2 t)+c_1 \sin (2 t)\right ) \\ \end{align*}