Solution of (2 y-x) d y/d x=1 is

Correct Answer is : x=2(y−1)+ce−yx=2(y-1)+{ce}^{-y}x=2(y−1)+ce−y, where c is the constant of integration

x=2(y-1)+{ce}^{-y}, where c is the constant of integration

x=2(y-1)+{ce}^{-x}, where c is the constant of integration

y=2(x-1)+{ce}^{-x}, where c is the constant of integration

y=2(x-1)+{ce}^{-y}, where c is the constant of integration

Correct Answer is : x=2(y−1)+ce−yx=2(y-1)+{ce}^{-y}x=2(y−1)+ce−y, where c is the constant of integration

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MHTCET 23 Apr 2025 Shift 2 Paper

Question : 122 of 150

Marks: +1, -0

Solution of

(2 y-x) \frac{d y}{d x}=1

Solution:

Concept:

Rewrite the given differential equation in the form

\frac{dx}{dy} + x = 2y

and solve using the integrating factor method.

Explanation:

Given

(2y - x)\frac{dy}{dx} = 1

Since

\frac{dy}{dx} \neq 0

, we can write

\frac{dx}{dy} = 2y - x

Rearranging gives

\frac{dx}{dy} + x = 2y

Here,

P = 1

and

Q = 2y

The integrating factor is

e^{\int 1\,dy} = e^y

The solution is

x \cdot e^y = \int 2y \cdot e^y\,dy + c

Using integration by parts:

\int 2y e^y\,dy = 2\left(y e^y - \int 1 \cdot e^y\,dy\right) = 2y e^y - 2e^y

Thus,

x e^y = 2e^y(y - 1) + c

Dividing by

e^y

, we get

x = 2(y - 1) + ce^{-y}

Answer:

\boxed{x = 2(y - 1) + ce^{-y}}

Hence, the correct option is A.

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