76.9.8 problem 8

Internal problem ID [17518]
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 : 8
Date solved : Tuesday, January 28, 2025 at 10:39:49 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-\frac {5 x \left (t \right )}{2}+\frac {3 y \left (t \right )}{2}\\ \frac {d}{d t}y \left (t \right )&=-\frac {3 x \left (t \right )}{2}+\frac {y \left (t \right )}{2} \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 3\\ y \left (0\right ) = -1 \end{align*}

✓ Solution by Maple

Time used: 0.068 (sec). Leaf size: 28

dsolve([diff(x(t),t) = -5/2*x(t)+3/2*y(t), diff(y(t),t) = -3/2*x(t)+1/2*y(t), x(0) = 3, y(0) = -1], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (-6 t +3\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (-18 t -3\right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 31

DSolve[{D[x[t],t]==-5/2*x[t]+3/2*y[t],D[y[t],t]==-3/2*x[t]+1/2*y[t]},{x[0]==3,y[0]==-1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-t} (3-6 t) \\ y(t)\to -e^{-t} (6 t+1) \\ \end{align*}