Find Parametric Equations for the Locus of Points?

d^2x/dt^2 = -dx/dt

Integrating both sides:

dx/dt = -x + C

Substituting dx/dt = 1, x = 1

1 = -1 + C

C = 2

So:

dx/dt = -x + 2

dx/dt + x = 2

Multiplying both sides by e^t:

dx/dt * e^t + x * e^t = 2e^t

dx/dt * e^t + x * d/dt(e^t) = 2e^t (as d/dt(e^t) = e^t)

d/dt(x * e^t) = 2e^t

Integrating both sides:

x * e^t = 2e^t + C

x = 2 + Ce^(-t)

Substituting x = 1, t = 0

1 = 2 + C * 1

C = -1

So:

x = 2 - e^(-t)

dy/dt = x

dy/dt = 2 - e^(-t)

Integrating both sides:

y = 2t + e^(-t)

So:

x = 2 - e^(-t)

y = 2t + e^(-t)

Hope this helps.

Curious curve, having the function

y = 2 - x - 2Log(2-x)

with the time-dependent parametric functions:

y = 2t + e^(-t)

x = 2 - e^(-t)

Edit: Northstar, Randy did a good job explaining. Thumbs up for him. I just guessed at it.

Edit 2: It was more fun for me to imagine a ball travelling as per parametric functions of t, where it simply zooms off nearly vertically as x -> 2 at the increasingly constant speed v-> 2, i.e. y ≈ 2t, after having started at v = √2.