7.23.3 problem 3

Internal problem ID [589]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.2 (Applications). Problems at page 345
Problem number : 3
Date solved : Monday, January 27, 2025 at 02:55:05 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=-3 x+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=-3 x+4 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 2 \end{align*}

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 33

dsolve([diff(x(t),t) = -3*x(t)+2*y(t), diff(y(t),t) = -3*x(t)+4*y(t), x(0) = 0, y(0) = 2], singsol=all)
 

\begin{align*} x \left (t \right ) &= -\frac {4 \,{\mathrm e}^{-2 t}}{5}+\frac {4 \,{\mathrm e}^{3 t}}{5} \\ y \left (t \right ) &= -\frac {2 \,{\mathrm e}^{-2 t}}{5}+\frac {12 \,{\mathrm e}^{3 t}}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 42

DSolve[{D[x[t],t]==-3*x[t]+2*y[t],D[y[t],t]==-3*x[t]+4*y[t]},{x[0]==0,y[0]==2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {4}{5} e^{-2 t} \left (e^{5 t}-1\right ) \\ y(t)\to \frac {2}{5} e^{-2 t} \left (6 e^{5 t}-1\right ) \\ \end{align*}