76.7.14 problem 14

Internal problem ID [17487]
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 : 14
Date solved : Tuesday, January 28, 2025 at 10:39:24 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 2\\ y \left (0\right ) = 5 \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 29

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

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{t}}{2}+\frac {3 \,{\mathrm e}^{-t}}{2} \\ y \left (t \right ) &= \frac {{\mathrm e}^{t}}{2}+\frac {9 \,{\mathrm e}^{-t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 40

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

\begin{align*} x(t)\to \frac {1}{2} e^{-t} \left (e^{2 t}+3\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (e^{2 t}+9\right ) \\ \end{align*}