76.9.11 problem 11

Internal problem ID [17521]
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 : 11
Date solved : Tuesday, January 28, 2025 at 10:39:52 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 28

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (3-5 t \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-\frac {t}{2}} \left (-25 t +5\right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 34

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

\begin{align*} x(t)\to e^{-t/2} (3-5 t) \\ y(t)\to e^{-t/2} (1-5 t) \\ \end{align*}