The integrating factor of y + frac{d}{dx}(xy) | vedclass

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Question

(C) Given the differential equation: $y + \frac{d}{dx}(xy) = x(\sin x + \log x)$.Expanding the derivative term: $y + y + x \frac{dy}{dx} = x(\sin x +

Vedclass · Accepted Answer

$x^2$

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(C) Given the differential equation: $y + \frac{d}{dx}(xy) = x(\sin x + \log x)$.Expanding the derivative term: $y + y + x \frac{dy}{dx} = x(\sin x + \log x)$.This simplifies to: $2y + x \frac{dy}{dx} = x(\sin x + \log x)$.Dividing the entire equation by $x$ (assuming $x 
eq 0$): $\frac{dy}{dx} + \frac{2}{x}y = \sin x + \log x$.This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P = \frac{2}{x}$ and $Q = \sin x + \log x$.The integrating factor $(IF)$ is given by $e^{\int P dx}$.$IF = e^{\int \frac{2}{x} dx} = e^{2 \log x} = e^{\log x^2} = x^2$.Therefore,the correct option is $C$.

The integrating factor of $y + \frac{d}{dx}(xy) = x(\sin x + \log x)$ is

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