The solution of the differential equation x² d y/d x-x y=1 is...

Correct Answer is : 2xy−2cx2+1=02 x y-2 c x²+1=02xy−2cx2+1=0

Correct Answer is : 2xy−2cx2+1=02 x y-2 c x²+1=02xy−2cx2+1=0

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MHTCET 15 Apr 2026 Shift 1 Paper

Question : 137 of 150

Marks: +1, -0

x^{2} \frac{d y}{d x}-x y=1

is...

Solution:

Concept:

Rewrite as a linear differential equation and use an integrating factor to solve.

Explanation:

Divide the given equation by

x^{2}

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

This is of the form

\frac{dy}{dx} + Py = Q

, where

P = -\frac{1}{x}

and

Q = \frac{1}{x^{2}}

Integrating factor:

\text{I.F.} = e^{\int -\frac{1}{x} dx} = e^{-\ln x} = \frac{1}{x}

Multiply the equation by

\frac{1}{x}

\frac{1}{x}\frac{dy}{dx} - \frac{y}{x^{2}} = \frac{1}{x^{3}}

The left side becomes

\frac{d}{dx}\left(\frac{y}{x}\right)

, so:

\frac{d}{dx}\left(\frac{y}{x}\right) = \frac{1}{x^{3}}

Integrate both sides:

\frac{y}{x} = \int x^{-3} dx = -\frac{1}{2x^{2}} + c

Multiply by

x

y = cx - \frac{1}{2x}

Multiply by

2x

and rearrange:

2xy - 2cx^{2} + 1 = 0

Answer:

2xy - 2cx^{2} + 1 = 0

Therefore, the correct option is

C

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