28.4.16 problem 7.16

Internal problem ID [4548]
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 : 7.16
Date solved : Monday, January 27, 2025 at 09:23:21 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )-5 x \left (t \right )+3 y&=2 \,{\mathrm e}^{3 t}\\ -x \left (t \right )+y^{\prime }-y&=5 \,{\mathrm e}^{-t} \end{align*}

✓ Solution by Maple

Time used: 0.133 (sec). Leaf size: 58

dsolve([diff(x(t),t)-5*x(t)+3*y(t)=2*exp(3*t),-x(t)+diff(y(t),t)-y(t)=5*exp(-t)],singsol=all)
 

\begin{align*} x &= c_{2} {\mathrm e}^{4 t}+{\mathrm e}^{2 t} c_{1} -4 \,{\mathrm e}^{3 t}-{\mathrm e}^{-t} \\ y &= -2 \,{\mathrm e}^{3 t}+\frac {c_{2} {\mathrm e}^{4 t}}{3}+{\mathrm e}^{2 t} c_{1} -2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.083 (sec). Leaf size: 99

DSolve[{D[x[t],t]-5*x[t]+3*y[t]==2*Exp[3*t],-x[t]+D[y[t],t]-y[t]==5*Exp[-t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{-t} \left (-8 e^{4 t}+(c_1-3 c_2) \left (-e^{3 t}\right )+3 (c_1-c_2) e^{5 t}-2\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (-4 e^{4 t}+(c_1-3 c_2) \left (-e^{3 t}\right )+(c_1-c_2) e^{5 t}-4\right ) \\ \end{align*}