75.30.2 problem 811

Internal problem ID [17264]
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 : Tuesday, January 28, 2025 at 09:58:35 AM
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*}

✓ Solution by Maple

Time used: 0.195 (sec). Leaf size: 32

dsolve([diff(x(t),t) = x(t)+y(t)-cos(t), diff(y(t),t) = -y(t)-2*x(t)+cos(t)+sin(t), x(0) = 1, y(0) = -2], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 31

DSolve[{D[x[t],t]==x[t]+y[t]-Cos[t],D[y[t],t]==-y[t]-2*x[t]+Cos[t]+Sin[t]},{x[0]==1,y[0]==-2},{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*}