64.16.6 problem 6

Internal problem ID [13609]
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 : 6
Date solved : Tuesday, January 28, 2025 at 05:53:49 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 )-3 x \left (t \right )-y \left (t \right )&=t\\ \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )-4 x \left (t \right )-y \left (t \right )&={\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.102 (sec). Leaf size: 41

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

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

✓ Solution by Mathematica

Time used: 0.710 (sec). Leaf size: 143

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

\begin{align*} x(t)\to e^{7 t} \left (\int _1^te^{-7 K[1]} \left (K[1]-e^{K[1]}\right )dK[1]+c_1\right ) \\ y(t)\to \frac {1}{6} e^t \left (-11 \left (e^{6 t}-1\right ) \int _1^te^{-7 K[1]} \left (K[1]-e^{K[1]}\right )dK[1]+6 \int _1^t\frac {1}{6} e^{-7 K[2]} \left (5 e^{6 K[2]} K[2]-11 K[2]+11 e^{K[2]}+e^{7 K[2]}\right )dK[2]-11 c_1 e^{6 t}+11 c_1+6 c_2\right ) \\ \end{align*}