The solution of the differential equation d y/d x= (x+y) is...

Correct Answer is : tan⁡(x+y2)=x+c ≤ft(x+y/2)=x+ctan(2x+y)=x+c

Correct Answer is : tan⁡(x+y2)=x+c ≤ft(x+y/2)=x+ctan(2x+y)=x+c

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MHTCET 19 Apr 2026 Shift 2 Paper

Question : 136 of 150

Marks: +1, -0

\frac{d y}{d x}=\cos (x+y)

is...

Solution:

Concept:

Use the substitution

v = x + y

to convert the given differential equation into a separable form.

Explanation:

Let

v = x + y

Then

\frac{dv}{dx} = 1 + \frac{dy}{dx} = 1 + \cos v

Therefore,

\frac{dx}{dv} = \frac{1}{1 + \cos v}

Using the identity

1 + \cos v = 2\cos^2\left(\frac{v}{2}\right)

, we get:

\frac{dx}{dv} = \frac{1}{2}\sec^2\left(\frac{v}{2}\right)

Integrating both sides with respect to

v

x = \tan\left(\frac{v}{2}\right) + C

Substitute back

v = x + y

x = \tan\left(\frac{x+y}{2}\right) + C

Rearranging gives

\tan\left(\frac{x+y}{2}\right) = x + C

, where

C = -c

(or simply write the constant as

c

This matches the required form.

Answer:

Option B:

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

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