76.6.4 problem 4

Internal problem ID [17459]
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 : 4
Date solved : Tuesday, January 28, 2025 at 10:39:02 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+2 y \left (t \right )+4\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+y \left (t \right )-3 \end{align*}

✓ Solution by Maple

Time used: 0.075 (sec). Leaf size: 44

dsolve([diff(x(t),t)=x(t)+2*y(t)+4,diff(y(t),t)=-2*x(t)+y(t)-3],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.190 (sec). Leaf size: 184

DSolve[{D[x[t],t]==x[t]+2*y[t]+4,D[y[t],t]==-2*x[t]+y[t]-3},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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