Internal problem ID [15511]
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 20. The method of elimination.
Exercises page 212
Problem number: 780.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )+\cos \left (t \right )\\ y^{\prime }\left (t \right )&=-4 y \left (t \right )+4 \cos \left (t \right )+3 x \left (t \right )-\sin \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 37
dsolve([4*diff(x(t),t)-diff(y(t),t)+3*x(t)=sin(t),diff(x(t),t)+y(t)=cos(t)],singsol=all)
\begin{align*} x \left (t \right ) &= \frac {c_{2} {\mathrm e}^{-3 t}}{3}+{\mathrm e}^{-t} c_{1} \\ y \left (t \right ) &= c_{2} {\mathrm e}^{-3 t}+{\mathrm e}^{-t} c_{1} +\cos \left (t \right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.218 (sec). Leaf size: 76
DSolve[{4*x'[t]-y'[t]+3*x[t]==Sin[t],x'[t]+y[t]==Cos[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{2} e^{-3 t} \left (c_1 \left (3 e^{2 t}-1\right )-c_2 \left (e^{2 t}-1\right )\right ) \\ y(t)\to \cos (t)+\frac {1}{2} e^{-3 t} \left (3 c_1 \left (e^{2 t}-1\right )-c_2 \left (e^{2 t}-3\right )\right ) \\ \end{align*}