28.4.21 problem 7.21

Internal problem ID [4553]
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 : 7.21
Date solved : Monday, January 27, 2025 at 09:23:25 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )-2 x \left (t \right )+y&=5 \,{\mathrm e}^{t} \cos \left (t \right )\\ x \left (t \right )+y^{\prime }-2 y&=10 \,{\mathrm e}^{t} \sin \left (t \right ) \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.145 (sec). Leaf size: 35

dsolve([diff(x(t),t)-2*x(t)+y(t) = 5*exp(t)*cos(t), x(t)+diff(y(t),t)-2*y(t) = 10*exp(t)*sin(t), x(0) = 0, y(0) = 0], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 30

DSolve[{D[x[t],t]-2*x[t]+y[t]==5*Exp[t]*Cos[t],x[t]+D[y[t],t]-2*y[t]==10*Exp[t]*Sin[t]},{x[0]==0,y[0]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -5 e^t (-\sin (t)+\cos (t)-1) \\ y(t)\to -5 e^t (\cos (t)-1) \\ \end{align*}