72.9.12 problem 26

Internal problem ID [14808]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.1. page 258
Problem number : 26
Date solved : Tuesday, January 28, 2025 at 07:17:02 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )-y\\ y^{\prime }&=2 x \left (t \right )-5 y \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 2\\ y \left (0\right ) = 3 \end{align*}

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 27

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 30

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

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