Problem 19 - Differential Equations and Linear Algebra

Question

Show that if \(\phi(x, y)\) is a potential function for \(M(x, y) d x+N(x, y) d y=0,\) then so is \(\phi(x, y)+c\) where \(c\) is an arbitrary constant. This shows that potential functions are only uniquely defined up to an additive constant.

Step-by-Step Solution

Answer

In short, given that \(\phi(x, y)\) is a potential function for \(M(x, y)dx + N(x, y)dy = 0\), we consider \(\psi(x, y) = \phi(x, y) + c\), where \(c\) is a constant. Since the partial derivatives of \(\psi(x, y)\) with respect to \(x\) and \(y\) are equal to \(M\) and \(N\), respectively: $$ \frac{\partial \psi}{\partial x} = M \quad\text{and}\quad \frac{\partial \psi}{\partial y} = N, $$ we conclude that \(\phi(x, y) + c\) is also a potential function for the given differential equation, showing that potential functions are only uniquely defined up to an additive constant.

Step 1: Find the partial derivatives of the given potential function

We know that \(\phi(x, y)\) is a potential function for the given differential equation \(M(x, y)dx + N(x, y)dy = 0\), so we have: $$ M = \frac{\partial \phi}{\partial x} \quad\text{and}\quad N = \frac{\partial \phi}{\partial y} $$

Step 2: Find the partial derivatives of \(\phi(x, y) + c\)

Let \(\psi(x, y) = \phi(x, y) + c\), where \(c\) is a constant. We need to find the partial derivatives of \(\psi(x, y)\) with respect to \(x\) and \(y\). Since \(c\) is a constant, its derivative will be \(0\). Therefore, $$ \frac{\partial \psi}{\partial x} = \frac{\partial(\phi(x, y) + c)}{\partial x} = \frac{\partial \phi}{\partial x} = M $$ and $$ \frac{\partial \psi}{\partial y} = \frac{\partial(\phi(x, y) + c)}{\partial y} = \frac{\partial \phi}{\partial y} = N $$

Step 3: Conclude that \(\phi(x, y) + c\) is also a potential function

From Step 2, we found that the partial derivatives of the new function \(\psi(x, y) = \phi(x, y) + c\) with respect to \(x\) and \(y\) are equal to \(M\) and \(N\), respectively. Therefore, \(\psi(x, y)\) also satisfies the conditions required for a potential function of \(M(x, y)dx + N(x, y)dy = 0\). This proves that \(\phi(x, y) + c\) is a potential function for the given differential equation, and potential functions are only uniquely defined up to an additive constant.

Key Concepts

Additive Constant

An additive constant is a number added to a function that does not change the behavior or the properties of that function, except for its overall value. When you combine it with the function, this constant does not affect the derivatives of the function because its derivative is zero.
In the context of a potential function, if we have a potential function \( \phi(x, y) \), adding an additive constant \( c \) results in a new function \( \psi(x, y) = \phi(x, y) + c \).
The key here is that this addition does not alter the essential characteristics of the potential function when we look at its derivatives.

• Additive constants only change the function value, not its derivative.
• They illustrate how different functions can be similar except for a shift in magnitude.

Such constants remind us that some properties, like potential functions, can be considered equivalent despite having different overall values.

Differential Equation

A differential equation is an equation that relates a function with its derivatives. It's a mathematical way to describe real-world phenomena where change is involved.
For instance, if you have the differential equation \( M(x, y)dx + N(x, y)dy = 0 \), this expresses a relationship between the rate of change of the potential function in the \( x \) and \( y \) directions.
This is central in fields like physics, engineering, and economics, where predicting behavior based on rates of change is crucial. Differentials equations allow us to:

• Model complex systems.
• Understand how variables influence each other.
• Predict future behaviors.

By finding a function that satisfies a given differential equation, we solve it and can understand the trajectory or state of the system being modeled.

Partial Derivatives

Partial derivatives allow us to explore the effect of changing one variable at a time in a multivariable function.
When you have a function of more than one variable, such as \( \phi(x, y) \), its partial derivates \( \frac{\partial \phi}{\partial x} \) and \( \frac{\partial \phi}{\partial y} \) help us understand how the function changes as \( x \) or \( y \) changes while keeping the other constant.
This is crucial in studying fields where functions involve several variables:

• Helps in understanding the behavior and form of the surface made by a function.
• Used to find the slope of the tangent to the surface in specified directions.
• Essential in calculating the gradient, critical points, and for optimization problems.

By analyzing these derivatives, we gain insightful information about the original function's shape, direction of steepest ascent, and stability.

Uniquely Defined

In mathematics, being uniquely defined means that there is one and only one solution to a problem, characteristic, or property of a function.
However, with potential functions, they are only uniquely defined up to an additive constant. This implies that you could have multiple potential functions differing only by a constant, yet all satisfy the same conditions of their derivatives.
In essence, while one might expect a completely exact solution, in many cases, several solutions are equivalent within a certain margin.

• It emphasizes flexibility in solutions.
• Highlights how different forms of the same function can essentially be equivalent.
• Promotes understanding that mathematical solutions might allow for variations.

Recognizing this helps in not only developing solutions but also comprehending the nature of the problem and its constraints.