73.27.5 problem 38.5

Internal problem ID [15761]
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 : 38.5
Date solved : Tuesday, January 28, 2025 at 08:06:46 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=5 x \left (t \right )+4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=8 x \left (t \right )+y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 9 \end{align*}

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 33

dsolve([diff(x(t),t) = 5*x(t)+4*y(t), diff(y(t),t) = 8*x(t)+y(t), x(0) = 0, y(0) = 9], singsol=all)
 

\begin{align*} x \left (t \right ) &= -3 \,{\mathrm e}^{-3 t}+3 \,{\mathrm e}^{9 t} \\ y \left (t \right ) &= 6 \,{\mathrm e}^{-3 t}+3 \,{\mathrm e}^{9 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 36

DSolve[{D[x[t],t]==5*x[t]+4*y[t],D[y[t],t]==8*x[t]+y[t]},{x[0]==0,y[0]==9},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to 3 e^{-3 t} \left (e^{12 t}-1\right ) \\ y(t)\to 3 e^{-3 t} \left (e^{12 t}+2\right ) \\ \end{align*}