65.14.3 problem 26.1 (iii)

Internal problem ID [13819]
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 (iii)
Date solved : Tuesday, January 28, 2025 at 06:04:43 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 41

dsolve([diff(x(t),t) = 2*x(t)+2*y(t), diff(y(t),t) = 6*x(t)+3*y(t)+exp(t), x(0) = 0, y(0) = 1], singsol=all)
 

\begin{align*} x \left (t \right ) &= -\frac {{\mathrm e}^{-t}}{7}+\frac {12 \,{\mathrm e}^{6 t}}{35}-\frac {{\mathrm e}^{t}}{5} \\ y \left (t \right ) &= \frac {3 \,{\mathrm e}^{-t}}{14}+\frac {24 \,{\mathrm e}^{6 t}}{35}+\frac {{\mathrm e}^{t}}{10} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.064 (sec). Leaf size: 58

DSolve[{D[x[t],t]==2*x[t]+2*y[t],D[y[t],t]==6*x[t]+3*y[t]+Exp[t]},{x[0]==0,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{35} e^{-t} \left (-7 e^{2 t}+12 e^{7 t}-5\right ) \\ y(t)\to \frac {1}{70} e^{-t} \left (7 e^{2 t}+48 e^{7 t}+15\right ) \\ \end{align*}