78.28.8 problem 1 (h)

Internal problem ID [18478]
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 (h)
Date solved : Tuesday, January 28, 2025 at 11:50:38 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: 0.019 (sec). Leaf size: 57

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: 59

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*}