72.10.17 problem 13 (a)

Internal problem ID [14828]
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 (a)
Date solved : Tuesday, January 28, 2025 at 07:17:18 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 ) = 1\\ y \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 33

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

\begin{align*} x \left (t \right ) &= \frac {2 \,{\mathrm e}^{-5 t}}{3}+\frac {{\mathrm e}^{-2 t}}{3} \\ y &= -\frac {2 \,{\mathrm e}^{-5 t}}{3}+\frac {2 \,{\mathrm e}^{-2 t}}{3} \\ \end{align*}

✓ Solution by Mathematica

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

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

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