50.28.2 problem 1(b)

Internal problem ID [8189]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 10. Systems of First-Order Equations. Section 10.3 Homogeneous Linear Systems with Constant Coefficients. Page 387
Problem number : 1(b)
Date solved : Monday, January 27, 2025 at 03:45:35 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.031 (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: 0.007 (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*}