74.22.11 problem 11

Internal problem ID [16658]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 6. Systems of Differential Equations. Exercises 6.1, page 282
Problem number : 11
Date solved : Tuesday, January 28, 2025 at 09:16:20 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 27

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 88

DSolve[{D[x[t],t]==y[t],D[y[t],t]==-x[t]+1},{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*}