28.4.17 problem 7.17

Internal problem ID [4549]
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 : 7.17
Date solved : Monday, January 27, 2025 at 09:23:22 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )-2 x \left (t \right )+y&=0\\ x \left (t \right )+y^{\prime }-2 y&=-5 \,{\mathrm e}^{t} \sin \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 53

dsolve([diff(x(t),t)-2*x(t)+y(t)=0,x(t)+diff(y(t),t)-2*y(t)=-5*exp(t)*sin(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.111 (sec). Leaf size: 82

DSolve[{D[x[t],t]-2*x[t]+y[t]==0,x[t]+D[y[t],t]-2*y[t]==-5*Exp[t]*Sin[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^t \left (-2 \sin (t)+4 \cos (t)+c_1 e^{2 t}-c_2 e^{2 t}+c_1+c_2\right ) \\ y(t)\to \frac {1}{2} e^t \left (2 \sin (t)+6 \cos (t)+c_1 \left (-e^{2 t}\right )+c_2 e^{2 t}+c_1+c_2\right ) \\ \end{align*}