63.20.5 problem 4

Internal problem ID [13227]
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 : 4
Date solved : Tuesday, January 28, 2025 at 05:12:43 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.068 (sec). Leaf size: 33

dsolve([diff(x(t),t) = x(t)-2*y(t), diff(y(t),t) = 3*x(t)-4*y(t), x(0) = 3, y(0) = 1], singsol=all)
 

\begin{align*} x \left (t \right ) &= 7 \,{\mathrm e}^{-t}-4 \,{\mathrm e}^{-2 t} \\ y &= 7 \,{\mathrm e}^{-t}-6 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 34

DSolve[{D[x[t],t]==x[t]-2*y[t],D[y[t],t]==3*x[t]-4*y[t]},{x[0]==3,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-2 t} \left (7 e^t-4\right ) \\ y(t)\to e^{-2 t} \left (7 e^t-6\right ) \\ \end{align*}