64.16.17 problem 17

Internal problem ID [13620]
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 : 17
Date solved : Tuesday, January 28, 2025 at 05:54:00 AM
CAS classification : system_of_ODEs

\begin{align*} 2 \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )-x \left (t \right )-y \left (t \right )&=1\\ \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )+2 x \left (t \right )-y \left (t \right )&=t \end{align*}

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 34

dsolve([2*diff(x(t),t)+diff(y(t),t)-x(t)-y(t)=1,diff(x(t),t)+diff(y(t),t)+2*x(t)-y(t)=t],singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {t}{3}-\frac {2}{9}+c_{2} {\mathrm e}^{3 t} \\ y \left (t \right ) &= -\frac {4}{9}-\frac {t}{3}-\frac {5 c_{2} {\mathrm e}^{3 t}}{2}+c_{1} {\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.183 (sec). Leaf size: 127

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

\begin{align*} x(t)\to e^{3 t} \left (\int _1^t-e^{-3 K[1]} (K[1]-1)dK[1]+c_1\right ) \\ y(t)\to \frac {1}{2} e^t \left (-5 \left (e^{2 t}-1\right ) \int _1^t-e^{-3 K[1]} (K[1]-1)dK[1]+2 \int _1^t\frac {1}{2} e^{-3 K[2]} \left (5 (K[2]-1)-e^{2 K[2]} (K[2]-3)\right )dK[2]-5 c_1 e^{2 t}+5 c_1+2 c_2\right ) \\ \end{align*}