76.7.11 problem 11

Internal problem ID [17484]
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 : 11
Date solved : Tuesday, January 28, 2025 at 10:39:22 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 )&=-5 x \left (t \right )+4 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.064 (sec). Leaf size: 34

dsolve([diff(x(t),t)=-2*x(t)+y(t),diff(y(t),t)=-5*x(t)+4*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{-t} \\ y \left (t \right ) &= 5 c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 72

DSolve[{D[x[t],t]==-2*x[t]+y[t],D[y[t],t]==-5*x[t]+4*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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