78.29.2 problem 1 (b)

Internal problem ID [18481]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 10. Systems of First Order Equations. Section 60. Critical Points and Stability for Linear Systems. Problems at page 539
Problem number : 1 (b)
Date solved : Tuesday, January 28, 2025 at 11:50:40 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-x \left (t \right )-2 y\\ y^{\prime }&=4 x \left (t \right )-5 y \end{align*}

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 56

dsolve([diff(x(t),t)=-x(t)-2*y(t),diff(y(t),t)=4*x(t)-5*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 62

DSolve[{D[x[t],t]==-x[t]-2*y[t],D[y[t],t]==4*x[t]-5*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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