76.8.7 problem 7

Internal problem ID [17496]
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 3.4 (Complex Eigenvalues). Problems at page 177
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 10:39:32 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 )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 4\\ y \left (0\right ) = -3 \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 44

dsolve([diff(x(t),t) = -x(t)-4*y(t), diff(y(t),t) = x(t)-y(t), x(0) = 4, y(0) = -3], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (4 \cos \left (2 t \right )+6 \sin \left (2 t \right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-t} \left (-4 \sin \left (2 t \right )+6 \cos \left (2 t \right )\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 46

DSolve[{D[x[t],t]==-x[t]-4*y[t],D[y[t],t]==x[t]-y[t]},{x[0]==4,y[0]==3},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-t} (4 \cos (2 t)-6 \sin (2 t)) \\ y(t)\to e^{-t} (2 \sin (2 t)+3 \cos (2 t)) \\ \end{align*}