64.16.7 problem 7

Internal problem ID [13610]
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 : 7
Date solved : Tuesday, January 28, 2025 at 05:53:50 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 65

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

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

✓ Solution by Mathematica

Time used: 7.401 (sec). Leaf size: 142

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

\begin{align*} x(t)\to \frac {1}{6} \left (-6 t+3 \left (c_1-\sqrt {6} c_2\right ) e^{-\sqrt {6} t}+3 \left (c_1+\sqrt {6} c_2\right ) e^{\sqrt {6} t}+1\right ) \\ y(t)\to \frac {1}{12} e^{-\sqrt {6} t} \left (2 e^{\sqrt {6} t} (t-1)-4 e^{\left (3+\sqrt {6}\right ) t}+\left (\sqrt {6} c_1+6 c_2\right ) e^{2 \sqrt {6} t}-\sqrt {6} c_1+6 c_2\right ) \\ \end{align*}