JAMIA 2021 — Mathematics PYQ

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Question

JAMIA 2021 — Mathematics PYQ

JAMIA | Mathematics | 2021

The solution of the differential equation $\frac{d y}{d x} e^{x - y} + x^{2} e^{- z}$ $y$ is:

Choose the correct answer:

Explanation

1. Given Differential Equation:

\frac{dy}{dx} = e^{x-y} + x^{2} e^{-y}

2. Simplify using the property of exponents ( $e^{a-b} = e^{a} \cdot e^{-b}$ ):

\frac{dy}{dx} = e^{x} e^{-y} + x^{2} e^{-y}

3. Factor out $e^{-y}$ on the right side:

\frac{dy}{dx} = e^{-y} (e^{x} + x^{2})

4. Separate the variables:

\frac{dy}{e^{-y}} = (e^{x} + x^{2}) dx

e^{y} dy = (e^{x} + x^{2}) dx

5. Integrate both sides:

\int e^{y} dy = \int (e^{x} + x^{2}) dx

6. Perform the integration:

e^{y} = e^{x} + \frac{x^{3}}{3} + C

Final Answer:

The solution is:

e^{y} - e^{x} = \frac{x^{3}}{3} + C