Problem 34 - Advanced Engineering Mathematics

Question

Verify that the given function satisfies Laplace's equation: $$ \frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0 $$ $$ z=e^{x^{2}-y^{2}} \cos 2 x y $$

Step-by-Step Solution

Answer

The function satisfies Laplace's equation.

Step 1: Compute the First Partial Derivatives

To begin, we must first find the first partial derivatives of the function \(z = e^{x^2 - y^2} \, \cos(2xy)\). Start by differentiating with respect to \(x\) and \(y\).### Partial Derivative with Respect to \(x\):Using the product rule and chain rule:\[\frac{\partial z}{\partial x} = \left( e^{x^2 - y^2} \right)' \cos(2xy) + e^{x^2 - y^2} \left( \cos(2xy) \right)' \]Computing derivatives:\[\frac{\partial z}{\partial x} = (2x)e^{x^2 - y^2} \cos(2xy) - 2y e^{x^2 - y^2} \sin(2xy)\]### Partial Derivative with Respect to \(y\):Using the product rule and chain rule:\[\frac{\partial z}{\partial y} = \left( e^{x^2 - y^2} \right)' \cos(2xy) + e^{x^2 - y^2} \left( \cos(2xy) \right)' \]Computing derivatives:\[\frac{\partial z}{\partial y} = (-2y)e^{x^2 - y^2} \cos(2xy) - 2x e^{x^2 - y^2} \sin(2xy)\]

Step 2: Compute the Second Partial Derivatives

Now, we need to find the second partial derivatives \(\frac{\partial^2 z}{\partial x^2}\) and \(\frac{\partial^2 z}{\partial y^2}\).### Second Partial Derivative with Respect to \(x\):Differentiate \(\frac{\partial z}{\partial x}\) with respect to \(x\):\[\frac{\partial^2 z}{\partial x^2} = \frac{\partial}{\partial x} \left( 2x e^{x^2 - y^2} \cos(2xy) - 2y e^{x^2 - y^2} \sin(2xy) \right)\]Applying the product and chain rules,\[\frac{\partial^2 z}{\partial x^2} = \left[ (2 + 4x^2) e^{x^2 - y^2} \cos(2xy) - 8xy e^{x^2 - y^2} \sin(2xy) \right] - 4y^2 e^{x^2 - y^2} \cos(2xy)\]### Second Partial Derivative with Respect to \(y\):Differentiate \(\frac{\partial z}{\partial y}\) with respect to \(y\):\[\frac{\partial^2 z}{\partial y^2} = \frac{\partial}{\partial y} \left( -2y e^{x^2 - y^2} \cos(2xy) - 2x e^{x^2 - y^2} \sin(2xy) \right)\]Applying the product and chain rules,\[\frac{\partial^2 z}{\partial y^2} = \left[ -(2 + 4y^2) e^{x^2 - y^2} \cos(2xy) + 8xy e^{x^2 - y^2} \sin(2xy) \right] - 4x^2 e^{x^2 - y^2} \cos(2xy)\]

Step 3: Verify Laplace's Equation

Add the second partial derivatives to see if their sum is zero:\[\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = [ (2 + 4x^2) e^{x^2 - y^2} \cos(2xy) - 8xy e^{x^2 - y^2} \sin(2xy) - 4y^2 e^{x^2 - y^2} \cos(2xy) ]\]\[+ [ -(2 + 4y^2) e^{x^2 - y^2} \cos(2xy) + 8xy e^{x^2 - y^2} \sin(2xy) - 4x^2 e^{x^2 - y^2} \cos(2xy) ]\]Combine similar terms:- Notice the terms \(8xy e^{x^2 - y^2} \sin(2xy)\) cancel each other.- Combine:\[(2 + 4x^2 - 4y^2) e^{x^2 - y^2} \cos(2xy) - (2 + 4y^2 + 4x^2) e^{x^2 - y^2} \cos(2xy) = 0\]Everything cancels out, verifying that the function satisfies Laplace's equation.

Key Concepts

Partial Derivatives

Partial derivatives allow us to explore how a multivariable function changes as we tweak one variable, keeping others constant. In our example, the task was to find the partial derivatives of the function \(z = e^{x^2 - y^2} \cos(2xy)\) with respect to both \(x\) and \(y\). By doing this, we examine how changes in \(x\) and \(y\) individually affect \(z\), helping us understand the function's behavior in different dimensions.
To compute these first partial derivatives, we utilize fundamental differentiation rules. The partial derivative with respect to \(x\) yields how \(z\) changes with a slight variation in \(x\), independent of \(y\). Similarly, the partial derivative with respect to \(y\) reveals how \(z\) varies as \(y\) changes, keeping \(x\) constant. This step sets the foundation for moving towards verifying Laplace's equation.

Chain Rule

The chain rule is an essential tool in calculus used for differentiating compositions of functions. It enables us to find the derivative of a composite function, a function comprised of other functions.
In the context of this problem, the chain rule is applied because our function \(z = e^{x^2 - y^2} \cos(2xy)\) involves compositions. The exponential \(e^{x^2 - y^2}\) and the trigonometric \(\cos(2xy)\) parts are both affected by changes in \(x\) and \(y\).
When differentiating \(e^{x^2 - y^2}\), the chain rule assists by first looking at \(x^2 - y^2\) as a separate derivation step before treating it as the exponent. This involves the derivative of the inner function (\(x^2 - y^2\)) used to modify the derivative of the outer function (\(e^u\) where \(u = x^2 - y^2\)). Similarly, for \(\cos(2xy)\), the rule ensures we calculate how each component within \(2xy\) affects the derivative.

Product Rule

The product rule is an essential differentiation technique used when dealing with products of two functions. It allows us to find the derivative of a product by multiplying each function by the derivative of the other and summing the results.
In this problem, the function \(z = e^{x^2 - y^2} \cos(2xy)\) is inherently a product of two functions. Therefore, we use the product rule along with the chain rule to compute the derivatives.
For both partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\), the product rule plays a critical role. By breaking down the problem, we differentiate \(e^{x^2 - y^2}\) and \(\cos(2xy)\) separately and then apply the rule to combine these results effectively. This combination process gives us a complete understanding of how our function \(z\) changes with respect to each variable.

Mathematical Verification

Mathematical verification provides the confidence that a function behaves as theoretically expected by classical equations—in this case, Laplace's equation. Laplace's equation, \(\frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}=0\), is fundamental in fields like physics and engineering to describe steady-state processes.
To verify Laplace's condition for our function \(z = e^{x^2 - y^2} \cos(2xy)\), we compute the second partial derivatives for \(x\) and \(y\) as outlined in the solution steps. After obtaining these, they are added together to check if the sum equals zero. Success in this step reassures us that the given function indeed satisfies Laplace's equation, confirming its steady-state profile.
This validation step reveals the elegance of mathematical analysis. By systematically applying rules and combining results, we see how the equation manifests in various fields, making sure every piece of the function aligns with theoretical principles.