65.14.4 problem 26.1 (iv)

Internal problem ID [13820]
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 (iv)
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 )&=5 x \left (t \right )-4 y \left (t \right )+{\mathrm e}^{3 t}\\ \frac {d}{d t}y \left (t \right )&=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: 0.056 (sec). Leaf size: 34

dsolve([diff(x(t),t) = 5*x(t)-4*y(t)+exp(3*t), diff(y(t),t) = x(t)+y(t), x(0) = 1, y(0) = -1], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (t^{2}+7 t +1\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{3 t} \left (t^{2}+6 t -2\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 39

DSolve[{D[x[t],t]==5*x[t]-4*y[t]+Exp[3*t],D[y[t],t]==x[t]+y[t]},{x[0]==1,y[0]==-1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{3 t} \left (t^2+7 t+1\right ) \\ y(t)\to \frac {1}{2} e^{3 t} \left (t^2+6 t-2\right ) \\ \end{align*}