72.12.11 problem 19

Internal problem ID [14858]
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 : 19
Date solved : Tuesday, January 28, 2025 at 07:17:46 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=4 x \left (t \right )+2 y\\ y^{\prime }&=2 x \left (t \right )+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: 23

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

\begin{align*} x \left (t \right ) &= \frac {1}{5}+\frac {4 \,{\mathrm e}^{5 t}}{5} \\ y &= \frac {2 \,{\mathrm e}^{5 t}}{5}-\frac {2}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 32

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

\begin{align*} x(t)\to \frac {1}{5} \left (4 e^{5 t}+1\right ) \\ y(t)\to \frac {2}{5} \left (e^{5 t}-1\right ) \\ \end{align*}