64.16.11 problem 11

Internal problem ID [13614]
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 : 11
Date solved : Tuesday, January 28, 2025 at 05:53:54 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 )+5 y \left (t \right )&=4 t\\ \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )+2 x \left (t \right )+2 y \left (t \right )&=2 \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 39

dsolve([2*diff(x(t),t)+diff(y(t),t)+x(t)+5*y(t)=4*t,diff(x(t),t)+diff(y(t),t)+2*x(t)+2*y(t)=2],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.116 (sec). Leaf size: 208

DSolve[{2*D[x[t],t]+D[y[t],t]+x[t]+5*y[t]==4*t,D[x[t],t]+D[y[t],t]+2*x[t]+2*y[t]==2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{-2 t} \left (\left (e^{6 t}+1\right ) \int _1^te^{-4 K[1]} \left (4 K[1]+e^{6 K[1]}-3\right )dK[1]-\left (e^{6 t}-1\right ) \int _1^te^{-4 K[2]} \left (-4 K[2]+e^{6 K[2]}+3\right )dK[2]+c_1 e^{6 t}-c_2 e^{6 t}+c_1+c_2\right ) \\ y(t)\to \frac {1}{2} e^{-2 t} \left (-\left (e^{6 t}-1\right ) \int _1^te^{-4 K[1]} \left (4 K[1]+e^{6 K[1]}-3\right )dK[1]+\left (e^{6 t}+1\right ) \int _1^te^{-4 K[2]} \left (-4 K[2]+e^{6 K[2]}+3\right )dK[2]+c_1 \left (-e^{6 t}\right )+c_2 e^{6 t}+c_1+c_2\right ) \\ \end{align*}