76.6.11 problem 11

Internal problem ID [17466]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 11
Date solved : Tuesday, January 28, 2025 at 10:39:07 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 )+2 \sin \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.071 (sec). Leaf size: 19

dsolve([diff(x(t),t) = y(t), diff(y(t),t) = -x(t)+2*sin(t), x(0) = 0, y(0) = 0], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 87

DSolve[{D[x[t],t]==y[t],D[y[t],t]==-x[t]+2*Sin[t]},{x[0]==0,y[0]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -\cos (t) \int _1^0-2 \sin ^2(K[1])dK[1]+\cos (t) \int _1^t-2 \sin ^2(K[1])dK[1]+\sin ^3(t) \\ y(t)\to \sin (t) \left (-\int _1^t-2 \sin ^2(K[1])dK[1]+\int _1^0-2 \sin ^2(K[1])dK[1]+\sin (t) \cos (t)\right ) \\ \end{align*}