28.4.13 problem 7.13

Internal problem ID [4545]
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 : 7.13
Date solved : Monday, January 27, 2025 at 09:23:18 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.171 (sec). Leaf size: 55

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

\begin{align*} x &= \sin \left (t \right ) c_{2} +\cos \left (t \right ) c_{1} +\frac {5 \sin \left (2 t \right )}{3} \\ y &= -\frac {c_{2} \cos \left (t \right )}{5}+\frac {c_{1} \sin \left (t \right )}{5}-\frac {2 \cos \left (2 t \right )}{3}-\frac {2 \sin \left (t \right ) c_{2}}{5}-\frac {2 \cos \left (t \right ) c_{1}}{5}-\frac {2 \sin \left (2 t \right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 67

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

\begin{align*} x(t)\to \frac {5}{3} \sin (2 t)+c_1 \cos (t)-2 c_1 \sin (t)-5 c_2 \sin (t) \\ y(t)\to -\frac {2}{3} \sin (2 t)-\frac {2}{3} \cos (2 t)+c_2 \cos (t)+c_1 \sin (t)+2 c_2 \sin (t) \\ \end{align*}