This is a case where an object is coupled to a spring, laying on a table. The object is moving, friction less, horizontally on the table. We assume the object is moving in an outer forice field which acts in the same direction as the object's motion. The motion is modeled by(adsbygoogle = window.adsbygoogle || []).push({});

y''(t) + y(t) = e[itex]^{-t}[/itex] (1)

where the right hand side is supposed to reflect the force field. (physics is not my field, so I'm not able to explain this in detail, I'm just a chemistry student with some extra maths).

This is a fairly simple problem, and I understand that the general solution can be written as a sum of the complimentary solution (of the homogeneous case) and one particular solution.

I've done the calculations, and found

y(t) = y[itex]_{C}[/itex](t) + y[itex]_{P}[/itex](t)

......= Acos(t) + Bsin(t) + 0,5e[itex]^{-t}[/itex]

However, I'm not sure if this is what I'm supposed to do.

I'm asked to

a) Find a particular solution y[itex]_{P}[/itex] on the form

y[itex]_{P}[/itex](t) = e[itex]^{-t}[/itex](Acos(t) + A sin(t)).

and

b) Find the general solution of the differential equation (1).

and finally

c) Assume the object from the start (y(0) = 0) is not moving (laying still). Find the objects position as a function of time in this case. What happens when t[itex]\rightarrow[/itex]∞?

What exactly am I supposed to do?

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# Solving a linear 2. order non-homogeneous differential equation

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