75.31.3 problem 817

Internal problem ID [17270]
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 23.2 The method of undetermined coefficients. Exercises page 239
Problem number : 817
Date solved : Tuesday, January 28, 2025 at 09:58:41 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 76

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

\begin{align*} x(t)\to -\sin (t) \int _1^t\cos (2 K[1])dK[1]-\cos ^3(t)+c_1 \cos (t)-c_2 \sin (t) \\ y(t)\to \cos (t) \int _1^t\cos (2 K[1])dK[1]-\sin (t) \cos ^2(t)+c_2 \cos (t)+c_1 \sin (t) \\ \end{align*}