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UPSC NDA I 2026 Mathematics Paper

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}

Question : 98 of 120

Marks: +1, -0

y(0)=0

Solution:

Concept:

Separate the variables and integrate using the given initial condition.

Explanation:

Given:

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

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

\Rightarrow e^{2y}\,dy=dx

Integrating both sides:

\frac{e^{2y}}{2}=x+C

Given

y(0)=0

\frac{1}{2}=0+C \Rightarrow C=\frac{1}{2}

Thus:

\frac{e^{2y}}{2}=x+\frac{1}{2}

\Rightarrow e^{2y}=2x+1

Taking natural logarithm:

2y=\ln(2x+1)

Answer:

2y=\ln(2x+1)

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