76.7.4 problem 4

Internal problem ID [17477]
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 3.3 (Homogeneous linear systems with constant coefficients). Problems at page 165
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 10:39:17 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.070 (sec). Leaf size: 34

dsolve([diff(x(t),t)=x(t)+y(t),diff(y(t),t)=4*x(t)-2*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} c_{1} +c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= -4 \,{\mathrm e}^{-3 t} c_{1} +c_{2} {\mathrm e}^{2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 71

DSolve[{D[x[t],t]==x[t]+y[t],D[y[t],t]==4*x[t]-2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{-3 t} \left (c_1 \left (4 e^{5 t}+1\right )+c_2 \left (e^{5 t}-1\right )\right ) \\ y(t)\to \frac {1}{5} e^{-3 t} \left (4 c_1 \left (e^{5 t}-1\right )+c_2 \left (e^{5 t}+4\right )\right ) \\ \end{align*}