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UPSC NDA I 2026 Mathematics Paper

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Question Numbers: 97-98

For the next two (02) items that follow:

Consider the differential equation

e^{x+y} \frac{d y}{d x}=e^{x-y}

:

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Question : 97 of 120

Marks: +1, -0

What is

\frac{d^{2} y}{d x^{2}}\left(\frac{d x}{d y}\right)^{2}

equal to ?

-2
-1
0
2

Solution:

Concept:

Solve the differential equation to express

y

in terms of

x

, then compute the required expression.

Explanation:

Given:

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

Cancelling

e^x

from both sides:

e^y\frac{dy}{dx}=e^{-y}

So,

e^{2y}\frac{dy}{dx}=1

\frac{d^2y}{dx^2} = \frac{d}{dx}\left(e^{-2y}\right) = -2e^{-2y}\frac{dy}{dx}

Since

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

, we get:

\frac{d^2y}{dx^2} = -2e^{-4y}

Also,

\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = e^{2y}

Therefore,

\frac{d^2y}{dx^2}\left(\frac{dx}{dy}\right)^2 = -2e^{-4y} \cdot e^{4y} = -2

Answer:

-2

, i.e., option A.

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