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MHTCET 23 Apr 2025 Shift 2 Paper

Question : 123 of 150

Marks: +1, -0

y+\frac{d}{d x}(x y)=x(\sin x+\log x)

Solution:

Concept:

Convert the given equation into the standard linear differential form

\frac{dy}{dx} + P(x)y = Q(x)

and then compute the integrating factor as

e^{\int P(x)\,dx}

Explanation:

Start with

y + \frac{d}{dx}(xy) = x(\sin x + \log x)

Expand the derivative:

\frac{d}{dx}(xy) = x\frac{dy}{dx} + y

So the equation becomes

y + x\frac{dy}{dx} + y = x(\sin x + \log x)

Simplifying:

x\frac{dy}{dx} + 2y = x(\sin x + \log x)

Divide throughout by

x

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

Here

P(x) = \frac{2}{x}

Integrating factor:

\text{I.F.} = e^{\int \frac{2}{x}dx} = e^{2\log x} = e^{\log x^2} = x^{2}

Answer:

The integrating factor is

x^{2}

, which corresponds to option C.

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