63.18.7 problem 3(c)

Internal problem ID [13213]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 190
Problem number : 3(c)
Date solved : Tuesday, January 28, 2025 at 05:12:30 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 71

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

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