Question

The general solution of \( \frac{dy}{dx} = e^{x - y} \) is:

• A. $e^y = e^x + c$
• B. $e^{-y} = e^x + c$
• C. $e^y = e^{-x} + c$
• D. $y = x + c$

Hint:

Remember: $e^{a-b} = e^a \cdot e^{-b}$. This allows you to separate the x and y variables easily.

Answer 1

The Correct Option is A

Step 1: Understand the Mathematical Concept
The given problem is a first-order ordinary differential equation: $\frac{dy}{dx} = e^{x-y}$. To solve this, we use the variable separable method. First, we use the laws of exponents to simplify the expression: $e^{x-y}$ can be rewritten as $\frac{e^x}{e^y}$. This allows us to group all terms containing '$y$' on one side and all terms containing '$x$' on the other.

Step 2: Differential Analysis & Integration
By cross-multiplying the terms to separate the variables, the equation becomes:
$e^y dy = e^x dx$

Now, we apply integration to both sides of the equation to find the general solution:
$\int e^y dy = \int e^x dx$
Since the integral of $e^u$ with respect to $u$ is simply $e^u$, the integration yields $e^y$ on the left and $e^x$ on the right.

Step 3: Conclusion & General Solution
After integrating, we must add an arbitrary constant ($c$) to represent the general solution of the differential equation:
$e^y = e^x + c$
This equation represents the relationship between $x$ and $y$ that satisfies the original derivative.

Final Answer: (A)