Differential Equations and Reality

Differential Equations
by Thomas Lee Abshier, ND
4/12/2025

Thomas Question: 

What is a differential equation? Is the solution to the equation the slope of the function at x? For example, what is the meaning of the differentials in Maxwell’s equations and the Thermodynamic equations? Does the solution to a differential equation give you the value of f(x) at x, or the slope of f(x) at x?

AI Answer:

A differential equation is a mathematical equation that relates a function

$f(x)$

(or a function of multiple variables, e.g.,

$f(x, y, t)$

to its derivatives. Derivatives describe how the function changes with respect to one or more variables, so a differential equation essentially describes the rate of change of a function and how it depends on the variables involved.

Let’s break this down in detail and address your questions about whether the solution to a differential equation gives the slope of the function or

$f(x)$

itself, with examples like Maxwell’s equations and thermodynamic partial differential equations.

---

1. What Is a Differential Equation?

A differential equation involves:

•  function (e.g., f(x)or
  $f(x,y,t)<span\; style="font-family:\; \text{'}Open\; Sans\text{'},\; Arial,\; sans-serif;">\; </span>$
• One or more derivatives of the function, such as
  $\frac{df}{dx}$ 

  or $\frac{\mathrm{\partial }f}{\mathrm{\partial }x}$, or higher-order derivatives like

  $\frac{d^{2}f}{d{x}^2}$

• A relationship between the function and its derivatives.

a. Ordinary Differential Equations (ODEs):

An ODE involves a function of a single variable and its derivatives. For example:

$$\frac{dy}{dx}=y$$

This is a simple ODE that states that the rate of change of

$y$

with respect to

$x$

is equal to

$y$

itself. The solution to this equation is

$y(x) = Ce^x$

, where

$C$

is a constant.

b. Partial Differential Equations (PDEs):

A PDE involves a function of multiple variables and its partial derivatives. For example:

$$\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2}$$

---

2. What Does the Solution to a Differential Equation Represent?

The solution to a differential equation is the function

$f(x)$

or

$f(x, y, t)$

etc. that satisfies the relationship described by the equation. It is not just the slope of the function at a given point but the entire function itself.

a. Derivatives Give the Slope or Rate of Change:

•  A derivative (e.g.,
  $\frac{dy}{dx}$ 

  ​gives the slope of the function $y(x)$at a particular point

  $x$

• A differential equation specifies how the slope (or higher-order rates of change) depends on $x$,
  $<span\; style="font-family:\; \text{'}Open\; Sans\text{'},\; Arial,\; sans-serif;">or\; other\; variable</span>$ 

b. The Solution is the Function $f(x)$

 

$<span\; style="color:\; \#666666;\; font-size:\; 14px;\; font-family:\; \text{'}Open\; Sans\text{'},\; Arial,\; sans-serif;">Solving\; the\; differential\; equation\; gives\; the\; full\; function</span>$

 

• 
  $f(x)<span\; style="font-family:\; \text{'}Open\; Sans\text{'},\; Arial,\; sans-serif;">,\; which\; describes\; the\; system’s\; behavior\; across\; all\; relevant\; values\; of\; x</span>$ 
• Once you know the solution
  $f(x)<span\; style="font-family:\; \text{'}Open\; Sans\text{'},\; Arial,\; sans-serif;">,\; you\; can\; calculate\; its\; slope\; at\; any\; point\; by\; taking\; its\; derivative.</span>$ 

Example:

For

$\frac{dy}{dx} = y$

, the solution

$y(x) = Ce^x$

gives the value of

$y$

at any

$x$

. The slope at any point is:

$$\frac{dy}{dx}=C{e}^x=y(x),<span\; style="font-family:\; \text{'}Open\; Sans\text{'},\; Arial,\; sans-serif;">\; </span>$$

which agrees with the original equation.

---

3. Maxwell’s Equations and Gradients

Maxwell’s equations describe how electric and magnetic fields behave and evolve. They are expressed as partial differential equations involving vector fields (e.g., electric field

$\mathbf{E}$

and magnetic field

$\mathbf{B}$

) and differential operators like gradients (

$\nabla$

), divergence (

$\nabla \cdot$

), and curl (

$\nabla \times$

).

Example: Faraday’s Law

One of Maxwell’s equations is:

$$\mathrm{\nabla }\times \mathbf{E}=-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\mathrm{.<span\; style="font-family:\; \text{'}Open\; Sans\text{'},\; Arial,\; sans-serif;"></span>}$$

• This equation describes how a time-varying magnetic field (B)
  induces a circulating electric field ( $\mathbf{E)}$ 
• The solution to this equation would be the full electric field
  $\mathbf{E}(\mathbf{r}, t)$ 

  $\mathbf{E}(\mathbf{r},t)$ as a function of position

  $\mathbf{r}$ 

  $\mathbf{r}$ and time

  $t$ 

  $t$, not just its slope or derivative.

Maxwell’s equations describe how the fields themselves (

$\mathbf{E}$

E and

$\mathbf{B}$

B) are related to their rates of change (gradients, curls, and time derivatives). The solution to these equations provides the fields themselves, not just their rates of change.

---

4. Thermodynamic Equations and Partial Differentials

In thermodynamics, partial differential equations frequently arise when modeling systems with multiple variables (e.g., temperature, pressure, entropy). For example:

Example: The Heat Equation

The heat equation is:

Here:

• 
  $u(x, t)$ 

  is the temperature at position x and time t.

• 
  $\mathrm{D<span\; style="font-family:\; \text{'}Open\; Sans\text{'},\; Arial,\; sans-serif;">is\; the\; thermal\; diffusivity.</span>}$ 

The solution to this equation is the temperature profile

$u(x, t)$

across space and time. The equation relates the rate of temperature change (

$\frac{\partial u}{\partial t}$

) to the curvature of the temperature profile (

$\frac{\partial^2 u}{\partial x^2}$

​), but the solution gives the actual temperature

$u(x, t)$

.

---

5. Summary of Key Points

• A differential equation relates a function to its derivatives, describing how the function changes over one or more variables.
• The solution to a differential equation is the function
  $f(x)$ 

  $\mathrm{,\; f}(x,y,t)<span\; style="font-family:\; \text{'}Open\; Sans\text{'},\; Arial,\; sans-serif;">,\; etc.)\; itself,\; not\; just\; its\; slope\; or\; derivative. </span>$

• Derivatives (e.g.,
  $\frac{dy}{dx}$ 

  ​) describe the slope or rate of change of the function at a given point.

• In examples like Maxwell’s equations or thermodynamic PDEs, the solution represents the physical quantity (e.g., electric field, temperature) as a function of space and time, with the differential equation describing how that quantity varies.

So, when solving a differential equation, we are not directly solving for the slope (

$\frac{df}{dx}$

​) but for the entire function

$f(x)$

, which inherently contains information about its slope, curvature, and higher-order behaviors as described by the derivatives.

 

---

Thomas Question:

You’re asking excellent and fundamental questions about the nature of differential equations, their purpose, and their role in describing systems. Let’s address your points step by step to clarify the value and purpose of writing and solving differential equations.

---

1. What Does $f(x) = \frac{df}{dx}$

 

 

 

 

 

Mean?

When you write

$f(x) = \frac{df}{dx}$

f(x)=dxdf​, you’re describing a functional relationship where the value of the function

$f(x)$

f(x) at any point

$x$

x is equal to the slope (rate of change) of

$f(x)$

f(x) at that same point.

• This is a differential equation because it involves the function
  $f(x)$ 

  $f(x)$ and its derivative

  $\frac{df}{dx}$ 

  dxdf​.

• The goal is to find the function
  $f(x)$ 

  $f(x)$ that satisfies this relationship.

Example: Solving $f(x) = \frac{df}{dx}$

 

 

 

 

 

 

 

 

f(x)=dxdf​

To solve

$f(x) = \frac{df}{dx}$

f(x)=dxdf​, we separate the variables and integrate:

where

$C$

C is a constant determined by boundary or initial conditions (e.g., the value of

$f(x)$

f(x) at a specific

$x$

x).

So yes, once solved,

$f(x)$

f(x) is simply a function that gives a value for each

$x$

x. There are no differentials left in the solution. The solution

$f(x) = Ce^x$

f(x)=Cex tells you both the value of the function at any

$x$

x and implicitly its slope, since

$\frac{df}{dx} = f(x)$

dxdf​=f(x) holds everywhere.

---

2. What Is the Value of Writing a Differential Equation?

Differential equations are powerful because they describe systems where the rate of change of a quantity is tied to the quantity itself or other variables. In many real-world systems, we don’t know the function

$f(x)$

f(x) directly, but we do know how it changes — and that is often enough to describe and understand the system.

The Value of Differential Equations

• A differential equation captures how a system evolves with respect to time, space, or other variables.
• Instead of directly giving the function
  $f(x)$ 

  $f(x)$, the equation tells us the rules that govern its behavior (e.g., how fast it grows, oscillates, or decays).

• By solving the equation, we construct the function
  $f(x)$ 

  $f(x)$ that satisfies these rules.

---

3. When Would One Use a Differential Equation?

Differential equations are used whenever the rate of change of a quantity is more important or more accessible than the quantity itself. They’re especially useful when:

1. The system evolves dynamically over time (e.g., velocity, population growth, heat transfer).
2. The relationship between a quantity and its rate of change is known (e.g., acceleration depends on velocity, or velocity depends on position).
3. Boundary or initial conditions are provided to specify a unique solution.

Examples of Systems Governed by Differential Equations

• Physics:
  • Newton’s second law:
    $F = ma$ 

    $F=ma$ becomes

    $m \frac{d^2x}{dt^2} = F(x, t)$ 

    mdt2d2x​=F(x,t), describing motion.

  • Maxwell’s equations: Describe the evolution of electric and magnetic fields.
• Biology:
  • Population growth:
    $\frac{dP}{dt} = rP(1 – \frac{P}{K})$ 

    dtdP​=rP(1−KP​) (logistic growth equation).

• Thermodynamics:
  • Heat conduction:
    $\frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2}$ 

    ∂t∂u​=D∂x2∂2u​.

• Finance:
  • Option pricing: Black-Scholes equation,
    $\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} – rV = 0$ 

    ∂t∂V​+21​σ2S2∂S2∂2V​+rS∂S∂V​−rV=0.

---

4. How Would You Declare a Useful Differential Equation?

To declare a useful differential equation, you must:

1. Identify the quantity of interest (e.g., position, velocity, temperature, population).
2. Determine how it changes with respect to time, space, or another variable.
3. Express this dependence mathematically (e.g., as a relationship between the quantity and its derivatives).

Formulating a Differential Equation

Let’s say you’re interested in the velocity of an object. You know that:

where

$a(t)$

a(t) is the acceleration. If you also know that the acceleration depends on time (e.g.,

$a(t) = -g$

a(t)=−g, where

$g$

g is the acceleration due to gravity), you can write:

This is a differential equation that describes how velocity changes over time. Solving it gives:

where

$C$

C is the initial velocity.

---

5. Why Isn’t $v = \frac{dx}{dt}$

 

 

 

 

 

 

 

 

v=dtdx​ a Differential Equation?

The equation

$v = \frac{dx}{dt}$

v=dtdx​ tells us what velocity is (the rate of change of position), but on its own, it doesn’t describe how velocity or position changes. It’s a definition, not a differential equation.

To make it a differential equation, you need to relate the rate of change (velocity) to something else. For example:

• If velocity depends on time, you could write:

   

  $$\frac{dx}{dt} = v(t),$$dtdx​=v(t),

  and then specify

  $v(t)$v(t) (e.g.,

  $v(t) = -gt + C$v(t)=−gt+C) to solve for position

  $x(t)$x(t).

• If velocity depends on position (e.g., a car braking), you could write:

   

  $$\frac{dx}{dt} = v(x),$$dtdx​=v(x),

  where

  $v(x)$v(x) is a known function of position.

---

6. The Key to Using Differential Equations

A differential equation is useful when it encodes how a system evolves over time, space, or some other variable. The solution provides the full behavior of the system, often including:

• The value of the quantity at every point (e.g., position
  $x(t)$ 

  $x(t)$).

• Its rate of change (e.g., velocity
  $\frac{dx}{dt}$ 

  dtdx​).

• Higher-order rates of change (e.g., acceleration
  $\frac{d^2x}{dt^2}$ 

  dt2d2x​).

---

7. Summary of Your Questions

• What is
  $f(x) = \frac{df}{dx}$ 

  f(x)=dxdf​?
  It’s a differential equation where the function

  $f(x)$ 

  $f(x)$ is equal to its own slope. Solving it gives the function

  $f(x)$ 

  $f(x)$, which you can evaluate at any

  $x$ 

  $x$.

• Why write differential equations?
  They describe systems where the rate of change is important (e.g., motion, population growth, heat transfer). They are particularly useful when we know how a quantity changes but not the quantity itself.
• How to use them effectively?
  To use a differential equation, relate the rate of change of a quantity to other variables or the quantity itself. Then solve the equation to recover the full behavior of the system.

Differential equations shine when the rate of change is the central feature of the system you’re studying. By solving them, you translate knowledge of “how things change” into a full understanding of “how things are.”

---

Thomas Question:

So, it sounds like the way we can think about differential equations is that they are a way of finding out a function (the behavior of some aspect of life) when we know how the function changes.

To make the differential equation accurately reflect reality, we would need to know the rate of the equation’s change that was equal to the actual behavior of the function.

Is this mainly useful in theoretical contexts, such as in research, when there is a curve of data, and then deriving the equation by doing curve fitting? (This seems awkward, and indirect, so that probably isn’t a place where using a differential equation would be useful.)

I guess the question is where differential equations would be useful. Obviously, you could use them as a theoretical illustration, to discuss how one quantity varied with another, but how are differential equations practical? Is it just by integrating them and getting the sum total of all the f(x1 to x2) in an interval that they are useful?

AI Answer:

Yes, you’re absolutely right that differential equations are a way of finding out a function (the behavior of some system) when we know how it changes — that is, when we know the rate of change of the function or how it depends on other variables. Let’s clarify the practicality of differential equations and their real-world applications, addressing your questions about their utility and relevance.

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1. How Do Differential Equations Reflect Reality?

To make a differential equation reflect reality, you need:

1. A clear understanding of how the system behaves, specifically how one quantity’s rate of change depends on itself or other variables.
2. Initial or boundary conditions, which give specific values for the function or its derivatives at certain points, allowing for a unique solution.

For example:

• In Newtonian mechanics, we know that the rate of change of velocity is proportional to the force acting on an object (
  $F = ma$ 

  $F=ma$).

• In population dynamics, we know that the growth rate of a population is proportional to the population size and environmental constraints (e.g., logistic growth).

Thus, differential equations are often derived directly from physical, biological, or economic principles. They encode the rules governing the system’s evolution, making them highly practical in modeling and predicting real-world behavior.

---

2. Are Differential Equations Mainly for Theoretical Contexts?

While differential equations are definitely used in theoretical research, their practical applications are vast and extend far beyond “curve fitting” or merely describing data. They are particularly valuable when:

a. The System’s Rules Are Known, but the Function Is Not

In many cases, we don’t know the function

$f(x)$

f(x) directly but can deduce how it changes based on physical laws or relationships.

Example 1: Projectile Motion

• If you drop an object, you know its acceleration is constant (
  $a = g$ 

  ).

• From
  $a = \frac{dv}{dt}$ 

  ​, you write

  $\frac{dv}{dt} = g$ 

  .

• Solving this gives
  $v(t) = gt + C$ 

  , where

  $C$ 

  is the initial velocity.

• To find position, use
  $\frac{dx}{dt} = v(t)$ 

  , which integrates to

  $x(t) = \frac{1}{2}gt^2 + C_1t + C_2$ 

  ​.
  This process models the object’s motion without direct data.

Example 2: Epidemic Modeling

• The spread of disease often follows the rule that the rate of new infections is proportional to the number of infected and susceptible individuals:

   

  $$\frac{dI}{dt} = \beta I S.$$.

• Solving this differential equation gives a model for the number of infected individuals over time, even when direct data is sparse.

b. Predicting Future Behavior

Differential equations are crucial for predictive models in science and engineering. For example:

• Weather modeling: The Navier-Stokes equations (differential equations for fluid flow) are used to predict atmospheric dynamics.
• Structural engineering: Differential equations describe how forces affect materials and structures over time.
• Economics: Differential equations model how prices or investments change over time in response to external factors.

c. When Direct Measurements Are Impractical

In many systems, you cannot directly measure the function

$f(x)$

f(x) at all points (e.g., temperature at every point in a metal bar, velocity at every moment in a trajectory). However, you can measure rates of change (e.g., heat flux, acceleration) and use differential equations to reconstruct the function.

---

3. Where Are Differential Equations Practical?

Differential equations are practical tools in a wide range of fields, not just theoretical exercises. Here are some specific practical applications:

a. Physics and Engineering

Differential equations describe nearly every physical system:

• Mechanics: Newton’s second law becomes a second-order ODE.
• Electrical Circuits: Kirchhoff’s laws lead to differential equations for current and voltage.
• Structural Engineering: Stress, strain, and deformation are described by PDEs like the wave equation.

b. Biology and Medicine

• Population Dynamics: Differential equations model population growth, predator-prey dynamics, and disease spread.
• Pharmacokinetics: Drug absorption, distribution, metabolism, and excretion are modeled by first-order ODEs.

c. Fluid Dynamics and Weather

• The Navier-Stokes equations, which describe fluid motion, are PDEs used to model weather, ocean currents, and aerodynamics.

d. Economics and Finance

• Differential equations model stock prices, interest rates, and economic growth over time. For example, the Black-Scholes equation predicts option prices.

e. Machine Learning

• Differential equations are used in neural networks and optimization problems to describe how weights change during training (gradient descent).

f. Environmental Science

• Models for heat diffusion, groundwater flow, and climate change are based on PDEs like the heat equation and advection-diffusion equations.

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4. Why Use Differential Equations Instead of Integrating Directly?

You asked whether the utility of differential equations lies in “integrating them to get the sum total over an interval.” While integration is part of solving differential equations, their value extends far beyond that.

a. Differential Equations Capture Relationships, Not Just Totals

A differential equation gives a rule for how something changes. Integration alone doesn’t give you this rule — it just sums up values over an interval.

For example:

• If you know the acceleration
  $a(t) = g$ 

  $a(t)=g$, integrating gives velocity, but only after you’ve formulated the differential equation

  $a = \frac{dv}{dt}$ 

  a=dtdv​.

• Differential equations describe how acceleration, velocity, and position are interrelated, giving a full picture of motion.

b. They Allow for General Solutions

Solving a differential equation gives a general solution — a function that works for all

$x$

x or

$t$

t (often with constants to adjust for initial conditions). This is more powerful than just computing specific totals in an interval.

c. They Are Essential for Complex Systems

In systems with multiple variables and rates of change (e.g., heat flow, population dynamics, or fluid motion), the relationships are too complex to simply “integrate.” Differential equations allow you to model and analyze these systems systematically.

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5. Summary of Practicality

• Differential equations are practical because they model how systems evolve over time or space.
• They are used to predict future behavior, describe relationships between variables, and solve for unknown functions when only rates of change are known.
• They are indispensable in engineering, physics, biology, economics, and more.
• While solving them often involves integration, their real value lies in describing relationships and rules of change, which integration alone cannot do.

When you write a differential equation, you essentially encode the laws of nature (or the system) into a mathematical framework, which can then be solved to understand and predict the system’s behavior.