64.16.5 problem 5

Internal problem ID [13608]
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 : 5
Date solved : Tuesday, January 28, 2025 at 05:53:48 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 )&={\mathrm e}^{-t}\\ \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 )&={\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 47

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

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

✓ Solution by Mathematica

Time used: 0.179 (sec). Leaf size: 230

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

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