28.4.11 problem 7.11

Internal problem ID [4543]
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 : 7.11
Date solved : Monday, January 27, 2025 at 09:23:16 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )-4 x \left (t \right )+3 y&=\sin \left (t \right )\\ -2 x \left (t \right )+y^{\prime }+y&=-2 \cos \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.262 (sec). Leaf size: 44

dsolve([diff(x(t),t)-4*x(t)+3*y(t)=sin(t),-2*x(t)+diff(y(t),t)+y(t)=-2*cos(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.274 (sec). Leaf size: 72

DSolve[{D[x[t],t]-4*x[t]+3*y[t]==Sin[t],-2*x[t]+D[y[t],t]+y[t]==-2*Cos[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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