76.7.7 problem 7

Internal problem ID [17480]
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 : 7
Date solved : Tuesday, January 28, 2025 at 10:39:19 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.072 (sec). Leaf size: 34

dsolve([diff(x(t),t)=5/4*x(t)+3/4*y(t),diff(y(t),t)=3/4*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}^{2 t} \\ y \left (t \right ) &= -{\mathrm e}^{\frac {t}{2}} c_{1} +c_{2} {\mathrm e}^{2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 80

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

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