64.16.15 problem 15

Internal problem ID [13618]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.1. Exercises page 277
Problem number : 15
Date solved : Tuesday, January 28, 2025 at 05:53:58 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 44

dsolve([2*diff(x(t),t)+4*diff(y(t),t)+x(t)-y(t)=3*exp(t),diff(x(t),t)+diff(y(t),t)+2*x(t)+2*y(t)=exp(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 76

DSolve[{2*D[x[t],t]+4*D[y[t],t]+x[t]-y[t]==3*Exp[t],D[x[t],t]+D[y[t],t]+2*x[t]+2*y[t]==Exp[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {3}{2} (c_1+c_2) e^{-2 t}-\frac {1}{2} e^t (2 t-1+c_1+3 c_2) \\ y(t)\to \frac {1}{6} e^t (6 t-1+3 c_1+9 c_2)-\frac {1}{2} (c_1+c_2) e^{-2 t} \\ \end{align*}