A family rather than an answer, and the constant that moves when you exponentiate.
Move the starting value, not the equation.
Every faint curve satisfies dP/dt = kP. Changing P(0) picks one out; changing k reshapes all of them at once.
Students who have only seen one worked solution think a differential equation has an answer. Seeing the sheaf first makes the initial condition obviously necessary rather than an extra step they forget.
ln|y| = x²/2 + c becomes y = Aex²/2, with A = ec. The additive constant becomes a multiplier.
A student who carries "+ c" through the exponential has an answer that does not satisfy the original equation, and the only way to notice is to substitute it back.
| dP/dt = kP | P = P₀ekt. With P₀ = 100 and k = 0.2, P(5) = 100e ≈ 271.8. |
| dy/dx = xy, y(0) = 1 | y = ex²/2, so y(2) = e² ≈ 7.389. |
| 1. Doubling culture | B, dN/dt = kN. |
| 2. P(5) | 271.8. |
| 3. y when x = 2 | 7.389. |
1 markForming the equation from the words, with letters defined.
1 markSeparating, with integral signs on both sides.
1 markIntegrating both sides correctly, including the constant.
1 markUsing the initial condition to find the constant.
| They give | What it means |
|---|---|
| dN/dt = 2 (Q1) | Read "doubles" as a constant rate. Ask what happens to a culture of a million against one of ten. |
| dN/dt = 2t (Q1) | Made the rate depend on the clock. Nothing in the sentence mentions elapsed time. |
| 200 (Q2) | Treated e as 2 because the context mentioned doubling. Worth catching; the exponent is 1 so it multiplies by e. |
| 2.718 (Q3) | Used exponent 1 rather than x²/2, which is 2 at x = 2. |
| 54.598 (Q3) | Forgot the division by 2 in x²/2. |
"Why only one constant?" Both integrations produce one, and the difference of two arbitrary constants is a single arbitrary constant. Say it once, clearly, or they write two and get confused.
"Does the modulus matter in ln|y|?" Here the population is positive so it is harmless, but keep it. It becomes A, positive or negative, after exponentiating.
"What if I cannot separate it?" Then it is not a question for this sub-topic. At this level everything that is set separates, and a slope field or Euler is the alternative tool.
| What is happening | |
|---|---|
| 1 | The family animation. One equation, many curves, and the measurement picks one. |
| 2 | Turning sentences into equations. Do several, including a cooling one with the minus sign. |
| 3 | Separation of variables, in full, with both integral signs written. |
| 4 | The constant becoming a multiplier, and checking the answer by substituting back. |
| 5 | Questions 1 to 3. |
| 6 | The exponential model recognised on sight. |
Do not let them write the answer as an exponential straight from dP/dt = kP without separating at least once. The recognition is useful; the method is what is marked.
Do not skip defining the variables. A question about cooling with T undefined is ambiguous about whether T is the temperature or the difference, and that ambiguity costs marks.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.