Solve: rac{dy}{dx} (x-y)=1

Correct Answer is : cot⁡(x−y2)=y+c ≤ft(x-y/2) = y + ccot(2x−y)=y+c

Correct Answer is : cot⁡(x−y2)=y+c ≤ft(x-y/2) = y + ccot(2x−y)=y+c

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UPSC NDA Math Model Paper 1

Question : 39 of 120

Marks: +1, -0

Solve:

rac{dy}{dx} \cos(x-y)=1

Solution:

Given,

rac{dy}{dx} \cos(x-y)=1

\Rightarrow \frac{dx}{dy} = \cos(x-y)

put

x-y=u

Differentiate w.

r

, to

y

, we get

\Rightarrow \frac{dx}{dy} - 1 = \frac{du}{dy}

\Rightarrow \frac{dx}{dy} = 1 + \frac{du}{dy}

Equation (1) becomes,

\Rightarrow 1 + \frac{du}{dy} = \cos u

\Rightarrow \frac{du}{dy} = \cos u - 1

\Rightarrow \frac{du}{\cos u - 1} = dy

variables are separated. Integrating on both side, we get

\Rightarrow \int \frac{du}{\cos u - 1} = \int dy

\Rightarrow \int \frac{du}{-2 \sin^{2} \frac{u}{2}} = \int dy

\Rightarrow -\frac{1}{2} \int \csc^{2}\left(\frac{u}{2}\right) du = \int dy

\Rightarrow \cot \left(\frac{u}{2}\right) = y + c

\cot \left(\frac{x-y}{2}\right) = y + c

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