75.30.3 problem 812

Internal problem ID [17265]
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 : 812
Date solved : Tuesday, January 28, 2025 at 09:58:37 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )+\tan \left (t \right )^{2}-1\\ \frac {d}{d t}y \left (t \right )&=\tan \left (t \right )-x \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 1.217 (sec). Leaf size: 29

dsolve([diff(x(t),t)=y(t)+tan(t)^2-1,diff(y(t),t)=tan(t)-x(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 100

DSolve[{D[x[t],t]==y[t]+Tan[t]^2-1,D[y[t],t]==Tan[t]-x[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \cos (t) \int _1^t-\cos (K[1])dK[1]+\sin (t) \int _1^t\sin (K[2]) \tan ^2(K[2])dK[2]+c_1 \cos (t)+c_2 \sin (t) \\ y(t)\to -\sin (t) \int _1^t-\cos (K[1])dK[1]+\cos (t) \int _1^t\sin (K[2]) \tan ^2(K[2])dK[2]+c_2 \cos (t)-c_1 \sin (t) \\ \end{align*}