Problem 44 - A Complete Resource Book in Mathematics for JEE Main

Question

Solution of the equation \(\frac{d y}{d x}=e^{x-y}\left(e^{x}-e^{y}\right)\) is (A) \(e^{y}=e^{x}-1+c e^{-e^{t}}\) (B) \(e^{y}=e^{x}-1+c e^{e^{e}}\) (C) \(e^{x}=e^{y}-1+c e^{-e^{\prime}}\) (D) None of these

Step-by-Step Solution

Answer

(D) None of these.

Step 1: Separate Variables

Start by rewriting the given differential equation \( \frac{d y}{d x} = e^{x-y}(e^{x} - e^{y}) \). Factoring the expression on the right can make it easier to separate variables. Notice that the equation becomes \( \frac{d y}{d x} = (e^x e^{-y})(e^x - e^y) \), which simplifies to \( \frac{d y}{d x} = e^{x}(e^{x-y} - 1) \).

Step 2: Integrate Both Sides

Rewrite the equation as \( \frac{d y}{d x} = e^{x} e^{x-y} - e^{x} \) and then separate the variables by rewriting it in terms of \( e^{y} \) and \( x \):\( \frac{d y}{e^{y}} + e^{-y} dy = (e^{x} - 1) dx \). Now integrate both sides separately.

Step 3: Integration Process

Integrate \( \int \frac{1}{e^{y}} dy \) and \( \int e^{-y} dy \) on the left side, and \( \int (e^x - 1) dx \) on the right. The integrals become \( - e^{-y} \) and \( e^{y} \) respectively for the left side, and \( e^x - x \) on the right side. Thus, the integrated equation is \( -e^{-y} + e^{y} = e^{x} - x + C \).

Step 4: Solve for e^y

Simplify the expression \( -e^{-y} + e^{y} \) and solve for \( e^y \): \( e^{-y}(e^{2y} - 1) = e^x - x + C \). Solving \( e^{2y} \) gives \( e^{2y} = e^x - x + C \, e^y \). Simplify further to get \( e^{y} = \sqrt{e^{x} - x + C} \).

Step 5: Check Solution Against Options

Now that we have solved for \( e^y \), compare the result with the given options: \( e^y = \sqrt{e^{x} - x + C} \). None of the given options (A, B, or C) match this expression. Therefore, the correct answer is (D) None of these.

Key Concepts

Separation of Variables

Separation of variables is a crucial technique in solving differential equations, especially when you have a product of functions of different variables. This technique simplifies the process by splitting the differential equation into two separate integrals, one involving only the dependent variable and the other involving the independent variable.

In our example, after rewriting the differential equation \( \frac{d y}{d x} = e^{x-y} (e^{x} - e^{y}) \), we follow the steps:

• Factor the expression on the right to simplify it into a form conducive to separation.
• The equation becomes \( \frac{d y}{d x} = (e^x e^{-y}) (e^x - e^y) = e^x (e^{x-y} - 1) \).
• Rewrite it as \( \frac{d y}{e^{y}} + e^{-y} dy = (e^{x} - 1) dx \).

Once the equation is in this form, you can integrate each part separately. This setup is essential, as improperly identifying this step can lead to errors in the solution.

Integration

Integration is used to solve the separate parts of a differential equation after the variables have been separated. This process transforms the equation into a solvable form where you can find a function that satisfies the original differential equation.

In this example, we have:

• For the left side: Integrate \( \int \frac{1}{e^{y}} dy \) and \( \int e^{-y} dy \).
• The first integral gives \( -e^{-y} \) and the second gives \( e^{y} \).
• For the right side: Integrate \( \int (e^x - 1) dx \).
• This integral gives \( e^x - x \).

Combining these results, you have the integrated form: \( -e^{-y} + e^{y} = e^{x} - x + C \).

Don't forget the constant of integration \( C \) which represents the family of curves that can satisfy the equation. It's essential to solving the differential equation completely.

Exponential Functions

Exponential functions are a recurring theme in differential equations because they are straightforward to integrate and differentiate. In our problem, the exponential function \( e^{x-y} \) occurs frequently, which requires careful handling of exponents and algebraic manipulation.

After integration, you obtain an equation with exponential terms such as \( -e^{-y} + e^{y} \). Solving for \( e^{y} \) involves algebraic manipulation and understanding the properties of exponential functions:

• Use exponential rules to simplify expressions, such as \( e^{a-b} = \frac{e^a}{e^b} \).
• Note that exponentials are always positive, impacting the range of solutions.
• Simplify complex expressions like \( e^{2y} \) by factoring out terms.

In this case, simplifying \( -e^{-y} + e^{y} \) and solving for \( e^{y} \) results in a more manageable equation: \( e^{y} = \sqrt{e^{x} - x + C} \).

Understanding exponential behaviors in terms of growth and decline is key, given that these functions model a wide array of real-world phenomena.