Pure 8.8 - Interpreting solutions of differential equations

You already know

For $k>0$, $e^{-kt}>0$ at every finite real $t$, and $e^{-kt}\to0$ as $t\to\infty$. A derivative describes a rate of change, with units of quantity per unit time. These facts are not printed as interpretation rules in the formula booklet.

Contextual interpretation

A contextual interpretation explains a solution's value, rate, restriction or limiting behaviour using the original quantity and units. A limit is a prediction of the model; it is not automatically an observed value or a value attained at a finite time.

Worked example 1: initial and limiting populations

Question. A population is modelled by $N(t)=20/(1+4e^{-0.3t})$, for $t\ge0$, with $N$ in thousands and $t$ in weeks. Interpret its initial and limiting values.

Plan. Substitute $t=0$, then let the exponential tend to zero.

Why multiply the values by $1000$? The formula counts thousands, not individual insects.

Answer. The model predicts $4000$ insects initially and a population approaching $20\,000$ insects in the long term.

Check. The denominator is greater than $1$ for every finite $t$, so $N(t)<20$: the limiting population is approached from below.

Worked example 2: a changed initial population

Question. Interpret the initial and limiting populations for $N(t)=20/(1+9e^{-0.3t})$, with the same units and $t\ge0$.

Plan. Evaluate the starting denominator and then its limit.

Why is the limit unchanged? The entire exponential term disappears in the limit, whatever its fixed coefficient.

Answer. The population starts at $2000$ insects and approaches $20\,000$ according to the model.

Check. Compared with Worked example 1, the denominator is larger at every finite time, so this population is smaller at each such time but has the same limit.

Watch out

Wrong: “The population is $20$ insects after a long time.” The formula measures thousands, and a limit need not be attained. Right: “The model predicts that the population approaches $20\,000$ insects.”

Worked example 3: decreasing towards a limit

Question. A reservoir-depth model gives $H(t)=0.4+1.1e^{-0.2t}$, with $H$ in metres, $t$ in minutes and $t\ge0$. Find its initial depth and limit, and determine whether it is increasing or decreasing.

Plan. Evaluate the endpoints in time and use the derivative's sign.

Why check the derivative? It establishes the direction throughout the interval rather than just comparing two sample depths.

Answer. The depth decreases from $1.5\,\mathrm m$ towards $0.4\,\mathrm m$. In a tank draining through a hole, the limit could represent the hole's height above the base, if that matches the setup.

Check. $H(t)-0.4=1.1e^{-0.2t}>0$, so the depth stays above its limit.

Worked example 4: increasing towards the same limit

Question. For $H(t)=0.4-0.2e^{-0.2t}$, in the same units, find the initial depth and limit and determine its direction of change.

Plan. Repeat the initial-value, limit and derivative checks.

Why does the direction reverse? Subtracting a positive term that is shrinking makes the total increase.

Answer. The depth increases from $0.2\,\mathrm m$ towards $0.4\,\mathrm m$, remaining below that limit at every finite time.

Check. For $t\ge0$, $0<e^{-0.2t}\le1$, so $0.2\le H(t)<0.4$.

Watch out

“Gradient tends to zero” does not imply “the quantity tends to a finite value”. For $r=(4t+25)^{2/3}$, $r'=(8/3)(4t+25)^{-1/3}\to0$, yet $r\to\infty$.

Mathematical and contextual restrictions

A mathematical restriction excludes undefined expressions, such as zero denominators or non-positive logarithm arguments. A contextual restriction excludes values that do not represent the situation, such as a negative depth or time before the model starts. The usable model interval must satisfy both, and its assumptions must remain reasonable.