72.12.2 problem 2

Internal problem ID [14849]
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 3.5 page 327
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 07:17:38 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+y\\ y^{\prime }&=-x \left (t \right )-2 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: 0.042 (sec). Leaf size: 105

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

\begin{align*} x \left (t \right ) &= \left (\frac {1}{2}+\frac {\sqrt {3}}{3}\right ) {\mathrm e}^{\sqrt {3}\, t}+\left (\frac {1}{2}-\frac {\sqrt {3}}{3}\right ) {\mathrm e}^{-\sqrt {3}\, t} \\ y &= \left (\frac {1}{2}+\frac {\sqrt {3}}{3}\right ) \sqrt {3}\, {\mathrm e}^{\sqrt {3}\, t}-\left (\frac {1}{2}-\frac {\sqrt {3}}{3}\right ) \sqrt {3}\, {\mathrm e}^{-\sqrt {3}\, t}-2 \left (\frac {1}{2}+\frac {\sqrt {3}}{3}\right ) {\mathrm e}^{\sqrt {3}\, t}-2 \left (\frac {1}{2}-\frac {\sqrt {3}}{3}\right ) {\mathrm e}^{-\sqrt {3}\, t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 82

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

\begin{align*} x(t)\to \frac {1}{6} e^{-\sqrt {3} t} \left (\left (3+2 \sqrt {3}\right ) e^{2 \sqrt {3} t}+3-2 \sqrt {3}\right ) \\ y(t)\to -\frac {e^{-\sqrt {3} t} \left (e^{2 \sqrt {3} t}-1\right )}{2 \sqrt {3}} \\ \end{align*}