65.14.7 problem 26.1 (vii)

Internal problem ID [13823]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 26, Explicit solutions of coupled linear systems. Exercises page 257
Problem number : 26.1 (vii)
Date solved : Tuesday, January 28, 2025 at 06:04:47 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=8 x \left (t \right )+14 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=7 x \left (t \right )+y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

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

dsolve([diff(x(t),t) = 8*x(t)+14*y(t), diff(y(t),t) = 7*x(t)+y(t), x(0) = 1, y(0) = 1], singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {4 \,{\mathrm e}^{15 t}}{3}-\frac {{\mathrm e}^{-6 t}}{3} \\ y \left (t \right ) &= \frac {2 \,{\mathrm e}^{15 t}}{3}+\frac {{\mathrm e}^{-6 t}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 44

DSolve[{D[x[t],t]==8*x[t]+14*y[t],D[y[t],t]==7*x[t]+y[t]},{x[0]==1,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{-6 t} \left (4 e^{21 t}-1\right ) \\ y(t)\to \frac {1}{3} e^{-6 t} \left (2 e^{21 t}+1\right ) \\ \end{align*}