50.29.6 problem 3(b)

Internal problem ID [8202]
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 A. Drill exercises. Page 400
Problem number : 3(b)
Date solved : Monday, January 27, 2025 at 03:45:45 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 32

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (c_{2} \cos \left (t \right )+c_{1} \sin \left (t \right )\right ) \\ y &= {\mathrm e}^{t} \left (\cos \left (t \right ) c_{1} -c_{2} \sin \left (t \right )\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 39

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

\begin{align*} x(t)\to e^t (c_1 \cos (t)+c_2 \sin (t)) \\ y(t)\to e^t (c_2 \cos (t)-c_1 \sin (t)) \\ \end{align*}