74.22.12 problem 12

Internal problem ID [16659]
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 : 12
Date solved : Tuesday, January 28, 2025 at 09:16:21 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.445 (sec). Leaf size: 38

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

\begin{align*} x \left (t \right ) &= c_{2} \sin \left (t \right )+\cos \left (t \right ) c_{1} -\frac {\sin \left (2 t \right )}{3} \\ y &= c_{2} \cos \left (t \right )-c_{1} \sin \left (t \right )-\frac {2 \cos \left (2 t \right )}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 86

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

\begin{align*} x(t)\to \sin (t) \int _1^t\cos (K[1]) \sin (2 K[1])dK[1]+c_2 \sin (t)+\cos (t) \left (-\frac {2 \sin ^3(t)}{3}+c_1\right ) \\ y(t)\to \cos (t) \int _1^t\cos (K[1]) \sin (2 K[1])dK[1]+\frac {2 \sin ^4(t)}{3}+c_2 \cos (t)-c_1 \sin (t) \\ \end{align*}