Question

Linear/Separable Differential Equations Question (Engineering First Year Calculus)?

I have no idea how to solve this question. If someone could do it or provide links to their answer, that would be great thanks :)

Answer 1

Starting with the first bit

Explanation:

Part (a)

Velocity

`(dv)/(dt) = -k v^n`, with `k gt 0`, and `n` as some constant

Separate:

`(dv)/(v^n) = -k \ dt`

Excluding `n = 1`, that solves as:

`(v^(1-n))/(1- n) = -kt + C`

With IV : `qquad v(0) = v_o`

`(v_o^(1-n))/(1- n) = C`

`(v^(1-n))/(1- n) = -kt + (v_o^(1-n))/(1- n)`

`v^(1-n) = v_o^(1-n) -k(1-n)t`

If `v(tau) = 0`:

`tau = 1/k * (v_o^(1-n))/(1- n)`

We expect that `tau gt 0`, or the model is meaningless. So:

`1/k * (v_o^(1-n))/(1- n) gt 0`

The givens:

• `k gt 0 implies 1/k gt 0`

• `v_0 gt 0 implies v_o^(1-n) gt 0`

`implies n lt 1`

Displacement

`(dv)/(dt) = v (dv)/(dx) = -k v^n, quad n lt 1`

`v^(1-n) dv = -k dx`

`(v^(2-n))/(2 - n) = - kx + C, qquad [n ne 2]`

With IV:

• `x(v_o) = 0`

`implies C = (v_o^(2-n))/(2 - n)`

`x =1/(k(2- n))( v_o^(2-n) - v^(2-n) )`

`v = 0 implies x =( v_o^(2-n) )/(k(2- n))`

The maximum distance depends upon the value of `v_0`. Eg representative values:

`x = {(n = -1, qquad v_0^3/(3k)), (n = 0, qquad v_0^2/(2k)), (n = 1, qquad v_0/k) :}`

[ Part (b), (c) - The rest of the question is answered in the above in one way or another]