Question

The general solution of the differential equation \[ \frac{dy}{dx}=e^{x-y}+x^2e^{-y} \] is

• A. \(e^{-y}=e^x+\frac{x^3}{3}+c\)
• B. \(e^y=e^x+\frac{x^3}{3}+c\)
• C. \(e^y=e^x+x^3+c\)
• D. \(e^y=e^x+c\)

Hint:

If \(e^{-y}\) appears in a differential equation, multiply by \(e^y\) to form \(\frac{d}{dx}(e^y)\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are required to find the general solution of a first-order differential equation.
The equation contains exponential terms with composite exponents. We can simplify this using laws of exponents to see if it's variable-separable.
Step 2: Key Formula or Approach:
Use the exponent rule \(e^{A+B} = e^A \cdot e^B\).
Factor out common terms to separate variables \(x\) and \(y\).
Integrate both sides independently.
Step 3: Detailed Explanation:

The given equation is: \(\frac{dy}{dx} = e^x \cdot e^{-y} + x^2 \cdot e^{-y}\).

Factor out the common term \(e^{-y}\) from the right-hand side:
\[ \frac{dy}{dx} = e^{-y} (e^x + x^2) \]

Now separate the variables by moving all \(y\) terms to one side and \(x\) terms to the other:
\[ \frac{1}{e^{-y}} dy = (e^x + x^2) dx \]
\[ e^y dy = (e^x + x^2) dx \]

Integrate both sides:
\[ \int e^y dy = \int (e^x + x^2) dx \]

Computing the integrals:
\[ e^y = e^x + \frac{x^3}{3} + c \]

Here, \(c\) is the arbitrary constant of integration.

Step 4: Final Answer:
The separation of variables leads to a direct integration of both sides, yielding the solution \(e^y = e^x + \frac{x^3}{3} + c\).