63.20.6 problem 5

Internal problem ID [13228]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 218
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 05:12:44 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 2\\ y \left (0\right ) = -1 \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 33

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

\begin{align*} x \left (t \right ) &= -\frac {3 \,{\mathrm e}^{2 t}}{2}+\frac {7 \,{\mathrm e}^{4 t}}{2} \\ y &= -\frac {9 \,{\mathrm e}^{2 t}}{2}+\frac {7 \,{\mathrm e}^{4 t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 44

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

\begin{align*} x(t)\to \frac {1}{2} e^{2 t} \left (7 e^{2 t}-3\right ) \\ y(t)\to \frac {1}{2} e^{2 t} \left (7 e^{2 t}-9\right ) \\ \end{align*}