76.9.2 problem 2

Internal problem ID [17512]
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 3.5 (Repeated Eigenvalues). Problems at page 188
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 10:39:45 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 {y \left (t \right )}{4} \end{align*}

✓ Solution by Maple

Time used: 0.070 (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)-1/4*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 57

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

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