72.9.10 problem 24

Internal problem ID [14806]
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 : 24
Date solved : Tuesday, January 28, 2025 at 07:17:00 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 y\\ y^{\prime }&=x \left (t \right )+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: 0.038 (sec). Leaf size: 33

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 44

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

\begin{align*} x(t)\to -\frac {2}{3} e^{-t} \left (2 e^{3 t}+1\right ) \\ y(t)\to \frac {1}{3} e^{-t} \left (1-4 e^{3 t}\right ) \\ \end{align*}