dP/dt = kP, P(0) = 50, P(2) = 150, find k Step by Step

Step by step

The question

Question: find k, if dP/dt = kP and P(0) = 50, P(2) = 150.

dPdt = kP

P(0) = 50, P(2) = 150

1 Read the equation

The rate of change is a number k times the amount P.

The more there is, the faster it changes.

the rate = k × P: the more P, the faster it changes

2 Separate and integrate

This is a separable equation.

Divide by P, times dt.

dPP = k dt

Integrate both sides, as in the separable lesson.

ln |P| = kt + C

Use e on both sides.

e^C is just a number: call it A.

P = Ae^(kt)

3 Use the starting value

Put in t = 0 and P = 50.

e⁰ = 1.

50 = Ae^(0)

So A = 50: the amount at the start.

A = 50

The formula: P = 50e^(kt).

P = 50e^(kt)

4 Find k

Now the second value: t = 2, P = 150.

150 = 50e^(2k)

Divide both sides by 50.

3 = e^(2k)

Careful!

k is inside a power.

To bring it down, take ln of both sides first.

Careful: not k = 15050 × 2 ✗

Take ln of both sides: ln and e cancel, so the power comes down.

ln 3 = 2k

Divide by 2: k is about 0.5493.

k = ln 32 ≈ 0.5493

5 Check

Let's check.

At t = 0: 50, the start.

Right!

P(0) = 50 × 1 = 50

With this k, at t = 2 the formula gives 150: the second value.

Right!

P(2) ≈ 50e^(0.5493 × 2) ≈ 150

Let's look back.

First: read the equation.

Then: separate and integrate.

Then: use the starting value.

Then: find k.

Last: check.

Answer