75.31.3 problem 817

Internal problem ID [17199]
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 : 817
Date solved : Monday, March 31, 2025 at 03:44:26 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.156 (sec). Leaf size: 28

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

\begin{align*} x \left (t \right ) &= c_1 \sin \left (t \right )+c_2 \cos \left (t \right ) \\ y \left (t \right ) &= -c_1 \cos \left (t \right )+c_2 \sin \left (t \right )+\sin \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 76

ode={D[x[t],t]==-y[t]+Sin[t],D[y[t],t]==x[t]+Cos[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.108 (sec). Leaf size: 39

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

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