69.30.2 problem 811

Internal problem ID [18551]
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. Methods of integrating nonhomogeneous linear systems with constant coefficients. Exercises page 234
Problem number : 811
Date solved : Thursday, October 02, 2025 at 03:14:49 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=-2 \\ \end{align*}

✓ Maple. Time used: 0.220 (sec). Leaf size: 32

ode:=[diff(x(t),t) = x(t)+y(t)-cos(t), diff(y(t),t) = -y(t)-2*x(t)+cos(t)+sin(t)]; 
ic:=[x(0) = 1, y(0) = -2]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 31

ode={D[x[t],t]==x[t]+y[t]-Cos[t],D[y[t],t]==-y[t]-2*x[t]+Cos[t]+Sin[t]}; 
ic={x[0]==1,y[0]==-2}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\sin (t)-t \cos (t)+\cos (t)\\ y(t)&\to t \sin (t)+(t-2) \cos (t) \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 110

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

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