28.4.20 problem 7.20

Internal problem ID [4552]
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 : 7.20
Date solved : Monday, January 27, 2025 at 09:23:24 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )-x \left (t \right )-2 y&=16 t \,{\mathrm e}^{t}\\ 2 x \left (t \right )-y^{\prime }-2 y&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 4\\ y \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.064 (sec). Leaf size: 49

dsolve([diff(x(t),t)-x(t)-2*y(t) = 16*t*exp(t), 2*x(t)-diff(y(t),t)-2*y(t) = 0, x(0) = 4, y(0) = 0], singsol=all)
 

\begin{align*} x &= 16 \,{\mathrm e}^{2 t}+{\mathrm e}^{-3 t}-12 t \,{\mathrm e}^{t}-13 \,{\mathrm e}^{t} \\ y &= 8 \,{\mathrm e}^{2 t}-2 \,{\mathrm e}^{-3 t}-8 t \,{\mathrm e}^{t}-6 \,{\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.195 (sec). Leaf size: 56

DSolve[{D[x[t],t]-x[t]-2*y[t]==16*t*Exp[t],2*x[t]-D[y[t],t]-2*y[t]==0},{x[0]==4,y[0]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -e^t (12 t+13)+e^{-3 t}+16 e^{2 t} \\ y(t)\to -2 e^t (4 t+3)-2 e^{-3 t}+8 e^{2 t} \\ \end{align*}