7.22.5 problem 15

Internal problem ID [580]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.1 (First order systems and applications). Problems at page 335
Problem number : 15
Date solved : Wednesday, February 05, 2025 at 03:45:51 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=\frac {y \left (t \right )}{2}\\ \frac {d}{d t}y \left (t \right )&=-8 x \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 35

dsolve([diff(x(t),t)=1/2*y(t),diff(y(t),t)=-8*x(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_1 \sin \left (2 t \right )+c_2 \cos \left (2 t \right ) \\ y \left (t \right ) &= 4 c_1 \cos \left (2 t \right )-4 c_2 \sin \left (2 t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 42

DSolve[{D[x[t],t]==1/2*y[t],D[y[t],t]==-8*x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to c_1 \cos (2 t)+\frac {1}{4} c_2 \sin (2 t) \\ y(t)\to c_2 \cos (2 t)-4 c_1 \sin (2 t) \\ \end{align*}