76.8.6 problem 6

Internal problem ID [17495]
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 : 6
Date solved : Tuesday, January 28, 2025 at 10:39:31 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 )\\ \frac {d}{d t}y \left (t \right )&=-5 x \left (t \right )-y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 49

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

\begin{align*} x \left (t \right ) &= c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right ) \\ y \left (t \right ) &= \frac {3 \cos \left (3 t \right ) c_{1}}{2}-\frac {3 \sin \left (3 t \right ) c_{2}}{2}-\frac {c_{1} \sin \left (3 t \right )}{2}-\frac {c_{2} \cos \left (3 t \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 54

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

\begin{align*} x(t)\to c_1 \cos (3 t)+\frac {1}{3} (c_1+2 c_2) \sin (3 t) \\ y(t)\to c_2 \cos (3 t)-\frac {1}{3} (5 c_1+c_2) \sin (3 t) \\ \end{align*}