dy/dt = -0.1y, y(0) = 50, find y(10) Step by Step

Step by step

The question

Question: find y(10), if dy/dt = −0.1y and y(0) = 50.

dydt = −0.1y

y(0) = 50

1 Read the equation

The rate of change is −0.1 times the amount y.

The more there is, the faster it shrinks.

k < 0: decay.

the rate = −0.1 × y: the more y, the faster it changes

2 Separate and integrate

This is a separable equation.

Divide by y, times dt.

dyy = −0.1 dt

Integrate both sides, as in the separable lesson.

ln |y| = −0.1t + C

Use e on both sides.

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

y = Ae^(−0.1t)

3 Use the starting value

Put in t = 0 and y = 50.

e⁰ = 1.

50 = Ae^(0)

So A = 50: the amount at the start.

A = 50

The formula: y = 50e^(−0.1t).

y = 50e^(−0.1t)

Careful!

This is not a steady rate.

The rate grows with y, so the formula has e, not a plus.

Careful: not y = 50 + −0.1 × 50t ✗

4 Answer the question

Put in t = 10.

y(10) = 50e^(−0.1 × 10)

The power: −1.

= 50e^(−1)

So y(10) ≈ 18.39.

≈ 18.39

5 Check

Let's check.

At t = 0: 50, the start.

Right!

y(0) = 50 × 1 = 50

And the derivative of the formula is −0.1 times y: the equation again.

Right!

y′ = 50 × −0.1e^(−0.1t) = −0.1 × y

Let's look back.

First: read the equation.

Then: separate and integrate.

Then: use the starting value.

Then: answer the question.

Last: check.

Answer