64.16.10 problem 10

Internal problem ID [13613]
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 : 10
Date solved : Tuesday, January 28, 2025 at 05:53:53 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-\frac {d}{d t}y \left (t \right )-2 x \left (t \right )+4 y \left (t \right )&=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 )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 39

dsolve([diff(x(t),t)-diff(y(t),t)-2*x(t)+4*y(t)=t,diff(x(t),t)+diff(y(t),t)-x(t)-y(t)=1],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{3 t}+c_{1} {\mathrm e}^{t}-\frac {t}{6}-\frac {13}{18} \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{3 t}+\frac {c_{1} {\mathrm e}^{t}}{3}-\frac {5}{18}+\frac {t}{6} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.117 (sec). Leaf size: 227

DSolve[{D[x[t],t]-D[y[t],t]-2*x[t]+4*y[t]==t,D[x[t],t]+D[y[t],t]-x[t]-y[t]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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