73.27.4 problem 38.4

Internal problem ID [15760]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 38. Systems of differential equations. A starting point. Additional Exercises. page 786
Problem number : 38.4
Date solved : Tuesday, January 28, 2025 at 08:06:45 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=5 x \left (t \right )-2 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 7\\ y \left (0\right ) = -7 \end{align*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 33

dsolve([diff(x(t),t) = x(t)+2*y(t), diff(y(t),t) = 5*x(t)-2*y(t), x(0) = 7, y(0) = -7], singsol=all)
 

\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{3 t}+4 \,{\mathrm e}^{-4 t} \\ y \left (t \right ) &= 3 \,{\mathrm e}^{3 t}-10 \,{\mathrm e}^{-4 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 44

DSolve[{D[x[t],t]==x[t]+2*y[t],D[y[t],t]==5*x[t]-2*y[t]},{x[0]==8,y[0]==-7},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {2}{7} e^{-4 t} \left (13 e^{7 t}+15\right ) \\ y(t)\to \frac {1}{7} e^{-4 t} \left (26 e^{7 t}-75\right ) \\ \end{align*}