Modelling With Differential Equations

Turning a described rate into an equation.

12 problems · about 21 minutes · FUN-7.A

What this lesson teaches

The student builds differential equations from verbal descriptions and interprets their solutions.

Translate the described rate into symbols before solving anything.
Proportional to the amount, to a difference, or inflow minus outflow are the three standard forms.
An initial condition selects the particular solution.

Straightforward practice. Get the method working first.

5 problems

In at 15 and out at 6 litres per minute. What is the net rate?
Answer 9
Why 9 L/min.
What does an initial condition do?
Answer It selects one solution from the family the equation describes.
Why It picks the particular solution.
In at 12 and out at 5 litres per minute. Net rate?
Answer 7
Why 12 − 5.
In at 3 and out at 8 litres per minute. Net rate?
Answer -5
Why 3 − 8.
A cup cools in a 25°C room. What temperature does it approach?
Answer 25
Why The surroundings.

The same ideas with more to keep track of.

3 problems

For a cooling object, is k positive or negative? 1 positive, 2 negative.
Answer 2
Why The temperature falls.
Does an initial condition select one solution from a family? 1 yes, 0 no.
Answer 1
Why That is its whole job.
"The rate is proportional to the amount present" gives dy/dt = k times what? 1 y, 2 t.
Answer 1
Why The amount present is y.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each description with its equation form.
Answer Rate proportional to the amount → dy by dt equals k y; Rate proportional to a difference from a constant → dT by dt equals k times (T minus Ts); Rate with a ceiling on growth → The logistic equation
Why Each phrase maps to one standard form.
In at 9 and out at 9 litres per minute. Net rate?
Answer 0
Why They balance.
The Room: A cup cools in a 20°C room. What temperature does it approach, in degrees?
Answer 20
Why 20°C.
The Tank: Water flows in at 10 litres per minute and out at 4. What is the net rate, in litres per minute?
Answer 6 L/min
Why 6 L/min.