A differential equation states how something changes. Modelling is translating a description of that change into symbols.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Proportional to the amount
"The rate is proportional to the amount present" is dy/dt = ky. It is the most common phrase in the topic.
Proportional to a difference
Newton cooling says the rate is proportional to the difference from the surroundings: dT/dt = k(T − Tₛ).
In and out
A tank problem sets the rate as inflow minus outflow. Each is expressed in the same units per unit time.
The initial condition
The equation gives a family of solutions. An initial condition selects the one that matches the situation.
Interpreting the answer
Report what the solution says about the situation, with units. A bare formula is only half the answer.
Turning a described rate into an equation
"The rate of change is proportional to the amount" becomes dy/dt = ky. "Proportional to the difference from 20" becomes dy/dt = k(y − 20). Translating the sentence is the whole modelling step.
Newton's law of cooling
An object cools at a rate proportional to the difference between its temperature and its surroundings. The solution approaches ambient temperature asymptotically, never quite reaching it — which matches experience.
The initial condition selects one solution
The general solution is a family; the initial condition picks the member. Exam questions almost always supply one and want the particular solution, with the constant evaluated.
Interpret the parameters
k is the rate constant, its sign tells growth from decay, and its size sets the timescale. Saying what each constant means in the situation is the final step of a complete modelling answer.
Step 2: Try It Yourself
Tap and try it out.
- The equationdy/dx = a·y
- Through(0, 2)
Every short line shows the slope the equation demands at that point. The curve simply follows them, and the marked point is the initial condition that picks it out of the family.
Step 3: Watch an Example
One step at a time.
Watch Rosa Model a Cooling Cup
A cup at 90°C cools in a 20°C room, with the rate proportional to the temperature difference.
- Step 1
The described rate is proportional to the difference from the room.
Step 4: Your Turn
Practice makes it stick.
The Room
Problem 1 of 2
A cup cools in a 20°C room. What temperature does it approach, in degrees?
The Tank
Problem 2 of 2
Water flows in at 10 litres per minute and out at 4. What is the net rate, in litres per minute?
L/min
Build the Equation
1 of 8
In at 12 and out at 5 litres per minute. Net rate?
2 of 8
In at 3 and out at 8 litres per minute. Net rate?
3 of 8
A cup cools in a 25°C room. What temperature does it approach?
4 of 8
For a cooling object, is k positive or negative? 1 positive, 2 negative.
5 of 8
Does an initial condition select one solution from a family? 1 yes, 0 no.
6 of 8
"The rate is proportional to the amount present" gives dy/dt = k times what? 1 y, 2 t.
7 of 8
Match each description with its equation form.
Tap a card on the left to start.
8 of 8
In at 9 and out at 9 litres per minute. Net rate?
Step 5: Quick Check
Show what you know.
Question 1 of 2
In at 15 and out at 6 litres per minute. What is the net rate?
Question 2 of 2
What does an initial condition do?
What You Learned
- 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.