▶️Differential Equations
Last updated
Last updated
Solve:
Just for reminder: the Inversely proportional means y = k/x where k is constant. Jump back to previous note: Proportional Relationship.
Assume the function of distance is S(t) = v · t.
So the speed must be the rate of change of distance, so the speed is v = S'(t)
Since the speed is inversely proportional to distance's square, so it means v = S'(t) = k/S²
Assume the amount of medication is M(t).
Now all the informations we have are:
M(0) = 150
M(13) = 150/2 = 75
M' = dM/dt = k · M because they're Proportional.
The problem is asking M(8) = ?.
So change a bit of M' to 1/M · dM = k · dt.
Take integral of each side to get ln(M) = k · t +C, and further M = C · eᵏᵗ
By introduce the initial condition, we get M(0) = 150 = C · e⁰ = C
By another information, we get M(13) = 150 · e¹³ᵏ = 75
And further, 13k = ln(1/2), so k = ln(0.5)/13
And now we get everything of the function M(t), let's solve for M(8)
`M(8) = 150 · e^(8 · ln(0.5)/13) ≃ 97.9
Solve:
Solve:
