75.31.1 problem 815

Internal problem ID [17268]
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 : 815
Date solved : Tuesday, January 28, 2025 at 09:58:39 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 )&=1-x \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 27

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 88

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

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