74.22.12 problem 12

Internal problem ID [16580]
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 : Thursday, March 13, 2025 at 08:23:40 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 )+\sin \left (2 t \right ) \end{align*}

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

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = -x(t)+sin(2*t)]; 
dsolve(ode);
 

\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*}

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

ode={D[x[t],t]==y[t],D[y[t],t]==-x[t]+Sin[2*t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.223 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(x(t) - sin(2*t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} - \frac {\sin ^{2}{\left (t \right )} \sin {\left (2 t \right )}}{3} - \frac {\sin {\left (2 t \right )} \cos ^{2}{\left (t \right )}}{3}, \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} - \frac {2 \sin ^{2}{\left (t \right )} \cos {\left (2 t \right )}}{3} - \frac {2 \cos ^{2}{\left (t \right )} \cos {\left (2 t \right )}}{3}\right ] \]