76.8.9 problem 9

Internal problem ID [17498]
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 : 9
Date solved : Tuesday, January 28, 2025 at 10:39:33 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-3 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 34

dsolve([diff(x(t),t) = x(t)-5*y(t), diff(y(t),t) = x(t)-3*y(t), x(0) = 1, y(0) = 1], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (-3 \sin \left (t \right )+\cos \left (t \right )\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (5 \cos \left (t \right )-5 \sin \left (t \right )\right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 34

DSolve[{D[x[t],t]==x[t]-5*y[t],D[y[t],t]==x[t]-3*y[t]},{x[0]==1,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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