76.7.8 problem 8

Internal problem ID [17481]
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 : 8
Date solved : Tuesday, January 28, 2025 at 10:39:20 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-\frac {3 x \left (t \right )}{4}-\frac {7 y \left (t \right )}{4}\\ \frac {d}{d t}y \left (t \right )&=\frac {x \left (t \right )}{4}+\frac {5 y \left (t \right )}{4} \end{align*}

✓ Solution by Maple

Time used: 0.068 (sec). Leaf size: 31

dsolve([diff(x(t),t)=-3/4*x(t)-7/4*y(t),diff(y(t),t)=1/4*x(t)+5/4*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} c_{1} +c_{2} {\mathrm e}^{t} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-\frac {t}{2}} c_{1}}{7}-c_{2} {\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 84

DSolve[{D[x[t],t]==-3/4*x[t]-7/4*y[t],D[y[t],t]==1/4*x[t]+5/4*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{6} e^{-t/2} \left (-\left (c_1 \left (e^{3 t/2}-7\right )\right )-7 c_2 \left (e^{3 t/2}-1\right )\right ) \\ y(t)\to \frac {1}{6} e^{-t/2} \left (c_1 \left (e^{3 t/2}-1\right )+c_2 \left (7 e^{3 t/2}-1\right )\right ) \\ \end{align*}