8.13.1 problem problem 3

Internal problem ID [922]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 7.2, Matrices and Linear systems. Page 417
Problem number : problem 3
Date solved : Wednesday, February 05, 2025 at 04:51:14 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 34

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

\begin{align*} x \left (t \right ) &= \sin \left (3 t \right ) c_1 +\cos \left (3 t \right ) c_2 \\ y \left (t \right ) &= -\cos \left (3 t \right ) c_1 +\sin \left (3 t \right ) c_2 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 68

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

\begin{align*} x(t)\to \frac {1}{2} e^{-3 t} \left (c_1 \left (e^{6 t}+1\right )+c_2 \left (e^{6 t}-1\right )\right ) \\ y(t)\to \frac {1}{2} e^{-3 t} \left (c_1 \left (e^{6 t}-1\right )+c_2 \left (e^{6 t}+1\right )\right ) \\ \end{align*}