75.30.5 problem 814

Internal problem ID [17267]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 23. Methods of integrating nonhomogeneous linear systems with constant coefficients. Exercises page 234
Problem number : 814
Date solved : Tuesday, January 28, 2025 at 09:58:38 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+\frac {1}{\cos \left (t \right )} \end{align*}

✓ Solution by Maple

Time used: 0.155 (sec). Leaf size: 47

dsolve([diff(x(t),t)=y(t),diff(y(t),t)=-x(t)+1/cos(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 43

DSolve[{D[x[t],t]==y[t],D[y[t],t]==-x[t]+1/Cos[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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