Problem 177 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ e^{x}-e^{y}=y-x $$

Step-by-Step Solution

Answer

After careful evaluation and manipulation of the original equation, it can be concluded that the solution is \(x = y\).

Step 1: Analyze and rearrange the equation

First, we can start by analyzing the given equation and then rearrange terms to get an equation which is easier to handle. Let's move \(e^{y}\) and \(x\) to the right-hand side, which results in: \(e^{x} - x = e^{y} + y\)

Step 2: Identify the form

Now we can notice something special about this. The left side of the equation looks like the derivative of the function \(e^{t} - t\) (where \(t\) is a placeholder variable). The derivative of this function is \(e^{t} - 1\), the equation could be rewritten as: \(f'(x) = f'(y)\) where \(f(t) = e^{t} - t\). Since the derivatives at these points are equal, and we know that \(f(t)\) is a one-to-one function that means it does not repeat the same values at different points, except at the point \(t = 1\), we conclude that the original x and y must be equal to each other.

Step 3: Confirming the solution

In order to solve for \(x\) and \(y\), we can plug the solution \(x = y\) back into the original equation to get confirmation. By substituting \(x\) for \(y\) in the original equation \(e^{x} - e^{y} = y - x\), we get 0 on both sides, confirming that \(x = y\) is the solution for this equation.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions that involve a constant base raised to a variable exponent. In the case of the original exercise, the term \(e^x\) and \(e^y\) are examples of exponential functions, where \(e\) is Euler's number, approximately equal to 2.71828.
Exponential functions have several unique characteristics:

• Rapid Growth or Decay: Exponential functions can grow very quickly or decay very fast depending on whether the exponent is positive or negative.
• Continuous and Differentiable: These functions are continuous everywhere and have a derivative that is also an exponential function.
• Natural Exponential Function: The function \(e^x\) is often referred to as the natural exponential functionbecause it has unique properties in calculus, particularly when differentiating or integrating.

In calculus, understanding how exponential functions behave is essential, as they frequently appear in various mathematical models of growth processes, from populations to financial investments. Moreover, they provide crucial building blocks for solving equations like the one in the original exercise.

Derivative

In calculus, a derivative represents the rate of change of a function concerning one of its variables. In simpler terms, it describes how a function's output changes as you make small changes to its input.
For exponential functions like \(e^t\), the derivative is particularly elegant: \[\frac{d}{dt}e^t = e^t\]This indicates that the rate of change of an exponential function is proportional to the function itself. The unique property, \(e^t\), remains unchanged even after differentiation.
In the step-by-step solution:

• The function \(f(t) = e^t - t\)is differentiated to obtain \(f'(t) = e^t - 1\).
• This derivative shows the rate of change of the function \(e^t - t\) at any point \(t\).
• By setting the derivatives \(f'(x)\) and \(f'(y)\) equal, implying where both change at the same rate, helps infer that when the derivative values match, x is equal to y.

The concept of derivatives is critical in calculus, enabling us to understand changes and estimate values in various fields, including physics and economics.

Equation Solving

Equation solving is a fundamental skill in mathematics that involves finding the values of variables that make an equation true. In this context, our equation \(e^{x} - e^{y} = y - x\)offers a scenario where rearranging and interpreting the equation helps in discovering solutions.
Here are steps typically taken to solve such equations:

• Rearrangement: Begin by manipulating the equation into a form that's more insightful. For instance: \(e^{x} - x = e^{y} + y\).
• Identifying Patterns: Notice the parts that resemble known functions—in this instance, recognizing the derivative form was key.
• Using Derivatives: Identify if differentiating can simplify the equation solving process.

The equation was tackled ingeniously through these observations:

• By equating derivatives, it was deduced that x and y must be equal, a critical insight to solving the problem effectively.
• Testing the solution by re-substituting ensures the solution's correctness.

Mastering equation solving is pivotal, allowing us to tackle a wide array of problems in both pure and applied mathematics. It is a skill that underpins much of advanced mathematics and science.