7.22.2 problem 12

Internal problem ID [577]
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 : 12
Date solved : Monday, January 27, 2025 at 02:55:03 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=y \left (t \right )\\ y^{\prime }\left (t \right )&=x \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

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

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

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