▶️Differential Equations

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Example

Solve:

Example

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

Example

Solve:

• 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