76.7.9 problem 9

Internal problem ID [17482]
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 : 9
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 {x \left (t \right )}{4}-\frac {3 y \left (t \right )}{4}\\ \frac {d}{d t}y \left (t \right )&=\frac {x \left (t \right )}{2}+y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.072 (sec). Leaf size: 35

dsolve([diff(x(t),t)=-1/4*x(t)-3/4*y(t),diff(y(t),t)=1/2*x(t)+y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 81

DSolve[{D[x[t],t]==-1/4*x[t]-3/4*y[t],D[y[t],t]==1/2*x[t]+y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -e^{t/4} \left (c_1 \left (2 e^{t/4}-3\right )+3 c_2 \left (e^{t/4}-1\right )\right ) \\ y(t)\to e^{t/4} \left (2 c_1 \left (e^{t/4}-1\right )+c_2 \left (3 e^{t/4}-2\right )\right ) \\ \end{align*}