28.4.19 problem 7.19

Internal problem ID [4551]
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 : 7.19
Date solved : Monday, January 27, 2025 at 09:23:23 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 1.738 (sec). Leaf size: 78

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

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

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 62

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

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