78.28.2 problem 1 (b)

Internal problem ID [18472]
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 56. Homogeneous Linear Systems with Constant Coefficients. Problems at page 505
Problem number : 1 (b)
Date solved : Tuesday, January 28, 2025 at 11:50:33 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 58

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 70

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

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