7 Differential Equations and Dynamic Models

Modeling Change with Differential Equations

A differential equation describes a quantity through its rate of change rather than specifying the quantity directly. If $y(t)$ is a quantity that varies with time $t$, then $\frac{dy}{dt}$ is its instantaneous rate of change. A general first-order model has the form

This equation says that the current rate depends on the time and state of the system. Applications include population change, radioactive decay, medication concentration, temperature, and motion.

An initial condition supplies the starting value, such as $y(0)=y_0$. Together, the differential equation and the initial condition form an initial-value problem. The desired solution must satisfy both the rate equation and the specified starting value.

A model is meaningful only under its assumptions. The variables, parameters, units, and time interval must correspond to the real system. For example, a population model should use a meaningful population scale and should not predict negative populations.

Takeaway: A differential equation gives the rule for change, while an initial condition identifies the particular trajectory that starts from the stated state.

Slope Fields and Equilibria

For a first-order equation

a slope field places a short segment at each point $(t,y)$. The segment at that point has slope $f(t,y)$, so a solution curve follows the local directions shown by the field.

Read the field qualitatively:

• Upward-tilting segments indicate $y'>0$, so solutions are increasing.

• Downward-tilting segments indicate $y'<0$, so solutions are decreasing.

• Horizontal segments indicate $y'=0$; a solution may have a local extremum there or may be constant.

For an autonomous equation, the slope depends only on the dependent variable:

Therefore, every horizontal line has the same slope pattern. Values satisfying $g(y)=0$ give equilibrium solutions. If the system starts at one of these values, it remains there.

Consider

The equilibrium values are $y=0$ and $y=4$. When $0<y<4$, the derivative is positive, so solutions increase. When $y>4$, the derivative is negative, so solutions decrease. When $y<0$, the derivative is also negative. Thus, $y=4$ attracts nearby positive solutions, while $y=0$ repels solutions that start just above it.

Takeaway: A slope field provides a visual and qualitative description of solution behavior before an explicit formula is found.

Solving Separable Equations

A separable differential equation can be written as

For values where $g(y)\neq 0$, separate the variables:

Then integrate both sides. A reliable procedure is:

1. Find constant solutions by solving $g(y)=0$.

2. Separate the variables.

3. Integrate both sides.

4. Solve explicitly for $y$, if possible.

5. Apply the initial condition.

The first step matters because dividing by $g(y)$ can remove constant solutions.

For example, consider

For $y\neq 0$, separation gives

After integration,

Exponentiating and incorporating the sign into the constant gives

The initial condition gives $C=3$, so the solution is

For $t>0$, this solution is increasing, and its rate of increase becomes faster because both $t$ and $y$ contribute to $y'=2ty$.

Takeaway: Separation converts a rate equation into two integrals, but constant solutions and the initial condition must be handled explicitly.

Exponential Growth and Decay

The exponential model assumes that the rate of change is proportional to the current amount:

Separating and solving gives

where $P_0=P(0)$. If $r>0$, the quantity grows; if $r<0$, it decays.

For growth, the doubling time is

For decay written with a positive decay constant $k$, so that $r=-k$, the half-life is

As an example, if a substance begins with $80$ milligrams and has decay constant $k=0.12$ per hour, then

After $5$ hours,

The model predicts an amount that approaches zero without reaching it in finite time. Its reliability depends on whether the proportional rate remains approximately constant. Unlimited exponential growth is often unrealistic because resources, space, competition, or other constraints eventually become important.

Takeaway: Exponential behavior is appropriate when the relative rate of change is approximately constant, but its assumptions should be checked over the intended time interval.

Logistic Growth and Long-Term Behavior

The logistic model incorporates a limiting level that restricts growth:

where $P(t)$ is the population, $r>0$ is the intrinsic growth rate, and $K>0$ is the carrying capacity. The factor $1-\frac{P}{K}$ reduces growth as $P$ approaches $K$.

With initial population $P(0)=P_0$, the solution is

The equilibrium solutions are $P=0$ and $P=K$. If $0<P<K$, then $P'>0$, so the population increases. If $P>K$, then $P'<0$, so it decreases toward $K$. For positive populations, $K$ is stable, while $P=0$ is unstable because a small positive population moves away from zero.

When $P$ is small compared with $K$, the correction factor is close to $1$, so logistic growth resembles exponential growth. Growth is fastest at $P=K/2$. At that point, the graph changes from accelerating growth to decelerating growth, producing the inflection point of the typical S-shaped curve. As $t$ becomes large,

Compared with the exponential model, the logistic model is more suitable when resources impose a meaningful upper limit. The choice between the models should be based on the system and the time interval, not only on which formula is easier to use.

Takeaway: Logistic growth begins approximately exponentially when the population is small, then slows and approaches the carrying capacity.

Interpreting and Evaluating a Model

A complete differential-equation analysis combines calculation with interpretation. Use the following checklist:

1. Identify each variable and its units, including the meanings of $t$, the dependent variable, and its derivative.

2. Find equilibria by setting the right-hand side equal to zero.

3. Determine where the derivative is positive or negative.

4. Use a slope field, sign analysis, or an explicit solution to describe short-term behavior.

5. Examine long-term behavior through limits, stable equilibria, or possible unbounded growth.

6. Check that the solution satisfies the initial condition.

7. Evaluate whether the parameters remain reasonable and whether the model's assumptions apply.

8. Respect the domain of the original quantity and the time interval for which the model was calibrated.

For the exponential equation $P'=rP$, growth continues without an upper bound when $r>0$. For the logistic equation $P'=rP(1-P/K)$, positive solutions approach $K$. The models can therefore agree when $P$ is small but differ substantially when $P$ becomes comparable to $K$.

A useful final interpretation should state what the solution means in the original context, not merely report a formula. Include the direction of change, the relevant equilibrium or limit, and the assumptions that make the prediction credible.

Takeaway: The strongest model analysis explains what the mathematics predicts, why it predicts it, and when that prediction should be trusted.

• equilibrium solution

• Stability