28.4.9 problem 7.9

Internal problem ID [4541]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.9
Date solved : Monday, January 27, 2025 at 09:23:15 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.167 (sec). Leaf size: 51

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

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

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 71

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

\begin{align*} x(t)\to e^{-t} \left (-t^2+c_1 \left (3 e^t-2\right )-3 c_2 \left (e^t-1\right )\right ) \\ y(t)\to e^{-t} \left (-t^2+t+2 c_1 \left (e^t-1\right )+c_2 \left (3-2 e^t\right )\right ) \\ \end{align*}