Question

Pls help:)
A particle moves along the x-axis, its displacement at time t being determined by the differential equation
\[\frac{d^2x}{dt^2}+2\frac{dx}{dt}+2x=cost\]
Find x in terms of t, and deduce that, when t is large, the motion of the particle is approximately simple harmonic of period \(2\pi\).

Answer 1

That is an inhomogeneous second order DE so first solve the homogeneous DE. I found this to be \[e^{-t}(asint+bcost)\]Then I would recommend the method of undetermined coefficients to find the particular solution. Assume the particular solution is of the form \[csint+dcost\] and solve the system of equations to find the coefficients. I found the particular solution to be \[cost\]The total solution is the sum of the homogeneous and particular solutions. To see the behavior of x(t) for large t, take the limit as t approaches infinite\[\lim_{t \rightarrow \infty} (cost +e^{-t}(asint+bcost))\]
which is \[cost\] and has a period of 2 pi.

Answer 2

Thnx a lot. that really helps:)