76.6.9 problem 9

Internal problem ID [17464]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 10:39:05 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.055 (sec). Leaf size: 30

dsolve([diff(x(t),t)=2*x(t)-y(t),diff(y(t),t)=3*x(t)-2*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t}+{\mathrm e}^{-t} c_{1} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{t}+3 \,{\mathrm e}^{-t} c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 73

DSolve[{D[x[t],t]==2*x[t]-y[t],D[y[t],t]==3*x[t]-2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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