76.8.8 problem 8

Internal problem ID [17497]
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 : 8
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 )&=2 x \left (t \right )-5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-2 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 3\\ y \left (0\right ) = 2 \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 25

dsolve([diff(x(t),t) = 2*x(t)-5*y(t), diff(y(t),t) = x(t)-2*y(t), x(0) = 3, y(0) = 2], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 26

DSolve[{D[x[t],t]==2*x[t]-5*y[t],D[y[t],t]==x[t]-2*y[t]},{x[0]==3,y[0]==2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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