The solution of the equation d y/d x=1/x+y+1 is

Correct Answer is : y=log⁡(x+y+2)+cy= (x+y+2)+cy=log(x+y+2)+c, where ccc is the constant of integration

x= (x+y+2)+c, where c is the constant of integration

x= (x+y-2)+c, where c is the constant of integration

y= (x+y+2)+c, where c is the constant of integration

y= (x+y-2)+c, where c is the constant of integration

Correct Answer is : y=log⁡(x+y+2)+cy= (x+y+2)+cy=log(x+y+2)+c, where ccc is the constant of integration

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MHTCET 5 May 2025 Shift 2 Paper

Question : 129 of 150

Marks: +1, -0

\frac{d y}{d x}=\frac{1}{x+y+1}

Solution:

Concept:

Use the substitution

v = x + y + 1

to convert the given differential equation into a separable form.

Explanation:

Let

v = x + y + 1

Differentiate with respect to

x

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

So,

\frac{dy}{dx} = \frac{dv}{dx} - 1

Substitute into the given equation:

\frac{dv}{dx} - 1 = \frac{1}{v}

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

Separating variables:

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

Write

\frac{v}{v+1}

1 - \frac{1}{v+1}

\int \left(1 - \frac{1}{v+1}\right) dv = \int dx

Integrating:

v - \log(v+1) = x + c_1

Substitute back

v = x + y + 1

x + y + 1 - \log(x + y + 2) = x + c_1

Simplify:

y = \log(x + y + 2) + c

, where

c = c_1 - 1

Answer:

Option C:

y = \log(x+y+2) + c

, where

c

is the constant of integration.

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