64.16.13 problem 13

Internal problem ID [13616]
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 : 13
Date solved : Tuesday, January 28, 2025 at 05:53:56 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 )&=t^{2}+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 t^{2}-2 t \end{align*}

✓ Solution by Maple

Time used: 0.093 (sec). Leaf size: 42

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

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

✓ Solution by Mathematica

Time used: 0.155 (sec). Leaf size: 211

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

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