64.16.16 problem 16

Internal problem ID [13619]
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 : 16
Date solved : Tuesday, January 28, 2025 at 05:53:59 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 )&=-2 t\\ \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )+x \left (t \right )-y \left (t \right )&=t^{2} \end{align*}

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 44

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

\begin{align*} x \left (t \right ) &= \frac {t^{2}}{2}+\frac {3 t}{2}+\frac {3}{4}+c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= \frac {15}{4}+\frac {3 t}{2}-\frac {t^{2}}{2}-3 c_{2} {\mathrm e}^{2 t}+c_{1} {\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.282 (sec). Leaf size: 115

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

\begin{align*} x(t)\to e^{2 t} \left (\int _1^t-e^{-2 K[1]} K[1] (K[1]+2)dK[1]+c_1\right ) \\ y(t)\to e^t \left (-3 \left (e^t-1\right ) \int _1^t-e^{-2 K[1]} K[1] (K[1]+2)dK[1]+\int _1^t-e^{-2 K[2]} K[2] \left (e^{K[2]} (K[2]+4)-3 (K[2]+2)\right )dK[2]-3 c_1 \left (e^t-1\right )+c_2\right ) \\ \end{align*}