74.22.9 problem 9

Internal problem ID [16656]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 6. Systems of Differential Equations. Exercises 6.1, page 282
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 09:16:19 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=-x-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=x+y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.035 (sec). Leaf size: 37

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

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 39

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

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