72.10.18 problem 13 (b)

Internal problem ID [14829]
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 3.2. page 277
Problem number : 13 (b)
Date solved : Tuesday, January 28, 2025 at 07:17:19 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 29

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

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

✓ Solution by Mathematica

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

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

\begin{align*} x(t)\to e^{-5 t}+e^{-2 t} \\ y(t)\to e^{-5 t} \left (2 e^{3 t}-1\right ) \\ \end{align*}