Solving the Equation 3ey/2 - 4e2y 30 Using Newton's Method and Mathematica

In this article, we will delve into solving a specific equation, 3ey/2 - 4e2y 30, using both graphical and numerical methods. We will also explore the solution using Mathematica, which provides a powerful symbolic and numerical computation environment.

Introduction to the Problem

Consider the equation 3ey/2 - 4e2y 30. We want to find the values of y that satisfy this equation. To start, we'll define a function f(y) based on the given equation and use it to transform the problem into a form amenable to solution:

f(y) 3ey/2 - 4e2y - 30

The problem can be restated as finding the root of this function, i.e., solving f(y) 0.

Graphical Solution

A graphical method can help us visualize the possible roots of the equation. By plotting the function f(y), we can identify the approximate value of the y-intercept which touches the x-axis. This allows us to narrow down the value of the real root.

From the plot, we observe a root with a value slightly less than 0.922.

Applying Newton's Method

To refine our estimate of the real root, we can use Newton's method. This iterative method is particularly effective for finding roots of functions. The formula for Newton's method is:

yn 1 yn - f(yn) / f'(yn)

First, we define the function and its derivative:

f(y) 3ey/2 - 4e2y - 30

f'(y) 3/2 ey/2 - 8e2y

The correction term is defined as:

newton[y] -30 3ey/2 - 4e2y / (3/2 ey/2 - 8e2y)

We start with an initial guess of y 0.91 and apply the method:

Iteration 1: 0.91 - (-30 3e0.91/2 - 4e2*0.91) / (3/2 e0.91/2 - 8e2*0.91) ≈ 0.921288

Iteration 2: 0.921288 - (-30 3e0.921288/2 - 4e2*0.921288) / (3/2 e0.921288/2 - 8e2*0.921288) ≈ 0.921167

Iteration 3: 0.921167 - (-30 3e0.921167/2 - 4e2*0.921167) / (3/2 e0.921167/2 - 8e2*0.921167) ≈ 0.921167

The value converges to six significant figures after three iterations.

Verification Using Mathematica

Mathematica provides a powerful tool to verify our solution. The Solve function can find all roots of the equation, both real and complex.

The output from Mathematica 11.3 is as follows:

{{Root -> 0.921167} {{Root -> 1.01087 - 3.05897i} {{Root -> 1.01087 3.05897i}}

These are the real and complex roots of the equation. To confirm, we can substitute each of these values into the original function f(y) and verify that the result is zero within the precision of digital calculations.

Conclusion

This article has demonstrated how to solve the equation 3ey/2 - 4e2y 30 using both graphical and numerical methods. We have seen the application of Newton's method and how Mathematica can be used to find and verify the solutions, including both real and complex roots. This approach not only provides a solution to the problem but also offers a comprehensive understanding of the underlying mathematical processes.