8.13.2 problem problem 4

Internal problem ID [923]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 7.2, Matrices and Linear systems. Page 417
Problem number : problem 4
Date solved : Wednesday, February 05, 2025 at 04:48:27 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 75

dsolve([diff(x(t),t)=3*x(t)-2*y(t),diff(y(t),t)=2*x(t)+y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \left (\sin \left (\sqrt {3}\, t \right ) c_1 +\cos \left (\sqrt {3}\, t \right ) c_2 \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (\sqrt {3}\, \sin \left (\sqrt {3}\, t \right ) c_2 -\sqrt {3}\, \cos \left (\sqrt {3}\, t \right ) c_1 +\sin \left (\sqrt {3}\, t \right ) c_1 +\cos \left (\sqrt {3}\, t \right ) c_2 \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 96

DSolve[{D[x[t],t]==3*x[t]-2*y[t],D[y[t],t]==2*x[t]+y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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