微分方程题目求解，最好有三问全过程

Question

如题所述

Answer 1

(1). du/dt + ru = A*Cos[wt]
e^(rt) * du/dt + e^(rt)*ru = A*e^(rt)*Cos[wt]
d(u*e^(rt))/dt = A*e^(rt)*Cos[wt]
u*e^(rt) - u(0) = Integrate[A*e^(rt)*Cos[wt], t]
= A*e^(rt) * (r*Cos[wt]+w*Sin[wt]) / (r^2+w^2) - A*r/(r^2+w^2)
so, u(t) = A * (r*Cos[wt]+w*Sin[wt]) / (r^2+w^2) + u(0)*e^(-rt) - A*r*e^(-rt)/(r^2+w^2).
(2). t->Infinity, u(t)~A*(r*Cos[wt]+w*Sin[wt]) / (r^2+w^2)
if u(0)=A*r/(r^2+w^2), then u(t) is periodic;(assume this solution is v)
if u(0)>A*r/(r^2+w^2), then u(t)>v and u(t)->v (t->Infinity)
if u(0)<A*r/(r^2+w^2), then u(t)<v and u(t)->v (t->infinity)
(3). u(0)=A*r/(r^2+w^2) if and only if u(t) is periodic.

Answer 2