Building Separable Models from Context

From a Verbal Description to a Differential Equation

A differential equation model begins with a statement about how a quantity changes. Phrases such as “the rate is proportional to the amount” or “the temperature changes in proportion to its difference from the surroundings” describe a relationship between a quantity and its derivative. The modeling task is to define the variables, translate the relationship precisely, and record the data that specify the situation.

A model is not just an equation with letters in it. Its variables need meanings and units, its parameters need interpretations, and its initial data need to match the information given. These details also help reveal whether the equation is separable.

Translate the Words Carefully

Suppose \(y\) is a quantity that depends on time \(t\). Its instantaneous rate of change is \(dy/dt\). If the rate is proportional to the amount \(y\), the mathematical meaning is that the ratio of the rate to the amount is constant, wherever \(y\ne 0\). Calling that constant \(k\) gives \(dy/dt=ky\). The equation itself also makes sense at \(y=0\); the ratio interpretation is simply not used there.

The coefficient \(k\) is not automatically positive. If \(y\) represents an increasing quantity and \(y>0\), then \(k>0\) describes growth. If it represents a decreasing quantity, a negative coefficient describes decay. In some contexts, it is clearer to write a positive constant and put the direction of change into a minus sign, as in a cooling or disappearance model.

Units provide a quick check. If \(y\) is measured in units \(U\) and \(t\) in units \(T\), then \(dy/dt\) has units \(U/T\). In \(dy/dt=ky\), \(k\) must therefore have units \(1/T\). The two sides of an equation must have the same units.

Other proportionality statements work similarly. If the rate of disappearance of a quantity is proportional to its square, the positive rate of disappearance is \(-dy/dt\), so the equation has the form \(-dy/dt=ky^2\), with \(k>0\). If a rate is proportional to a difference, first define that difference and then attend to its sign.

Why These Models Are Separable

The structural reason is direct. If a contextual statement yields a rate of the form \(g(t)h(y)\), then the differential equation is separable by definition. On an interval where \(h(y)\ne 0\), its variables can be placed on opposite sides as \(dy/h(y)=g(t)\,dt\). This rearrangement is a way of recognizing the structure; it does not by itself determine a solution.

The condition matters: division by \(h(y)\) is valid only where \(h(y)\ne 0\). The original differential equation remains the model, including at values where that factor is zero. The modeling step should therefore identify the equation first, rather than immediately divide by a quantity that might vanish.

Worked Example: Population Growth Proportional to Population

Temperature Difference and the Direction of Change

For an object cooling in surroundings held at a fixed temperature, the relevant difference is not the object’s temperature by itself. It is the object’s temperature relative to the surroundings. Write \(T(t)\) for the object’s temperature and \(T_a\) for the constant ambient temperature. If the object is warmer than its surroundings, \(T-T_a\) is positive, while its temperature is decreasing. A positive constant \(k\) therefore gives the model \(dT/dt=-k(T-T_a)\).

The minus sign expresses the direction of change for the case \(T>T_a\). If the object is cooler than the surroundings, then \(T-T_a<0\), so the same model predicts \(dT/dt>0\): its temperature rises. Thus the equation describes change toward the ambient temperature, not just cooling from above.

Worked Example: Cooling Toward a Constant Ambient Temperature

Worked Example: A Rate Proportional to Concentration Squared

Initial Conditions, Measurements, and Model Assumptions

An initial condition gives the value of the dependent variable at a particular independent-variable value. For a quantity \(y(t)\), it has the form \(y(t_0)=y_0\). A measured derivative, such as a growth rate at \(t_0\), is not itself an initial condition. It may instead provide information for estimating an unknown parameter, as in the examples.

A model may also include known external quantities, such as the ambient temperature, or an assumed constant, such as the proportionality coefficient. Keep their roles distinct:

These equations also rely on contextual assumptions. For example, a constant proportionality factor assumes that the rate-to-quantity relationship does not change over the modeled interval. A fixed-ambient-temperature cooling model assumes the surroundings remain at the stated temperature. Such assumptions are not guaranteed by calculus; they are part of the mathematical description of the situation.

A Common Sign Mistake

In the cooling example, a tempting but incorrect translation is \(dT/dt=k(T-20)\) with \(k>0\). At the initial temperature \(85^\circ\text{C}\), this would predict \(dT/dt=65k>0\), meaning the drink warms. That contradicts both the description and the measured negative rate.

The correction is to write the direction into the rate law: \(dT/dt=-k(T-20)\) for positive \(k\). With \(k=0.2\), substitution gives \(-0.2(85-20)=-13\), so the model predicts cooling at the stated rate. Another common error is to use \(T\) rather than \(T-T_a\): that would make the rate depend on the Celsius zero point, even though the described effect depends on the difference between the drink and its surroundings.

Check the Model Before Using It

Before proceeding from a verbal situation to later calculations, check that the model preserves the information in the description. A useful sequence is to define each variable and unit, write the rate as a derivative, translate proportionality, determine the sign from the direction of change, and state the initial condition. Then check units and substitute any measured values into the equation.