The general solution of the differential equa | vedclass

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(D) Step $1$: Divide the equation by $x \sin x$ to get the standard linear form $\frac{dy}{dx} + P(x)y = Q(x)$.$\frac{dy}{dx} + \frac{x \cos x + \sin

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$xy \sin x + \cos x = c$

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(D) Step $1$: Divide the equation by $x \sin x$ to get the standard linear form $\frac{dy}{dx} + P(x)y = Q(x)$.$\frac{dy}{dx} + \frac{x \cos x + \sin x}{x \sin x} y = \frac{1}{x}$.Step $2$: Identify $P(x) = \frac{x \cos x + \sin x}{x \sin x} = \cot x + \frac{1}{x}$.Step $3$: Find the integrating factor $IF = e^{\int P(x) dx} = e^{\int (\cot x + \frac{1}{x}) dx} = e^{\ln |\sin x| + \ln |x|} = e^{\ln |x \sin x|} = x \sin x$.Step $4$: The solution is $y \cdot IF = \int Q(x) \cdot IF dx$.$y(x \sin x) = \int \frac{1}{x} (x \sin x) dx = \int \sin x dx$.$xy \sin x = -\cos x + c$.$xy \sin x + \cos x = c$.

The general solution of the differential equation $x \sin x \frac{dy}{dx} + (x \cos x + \sin x)y = \sin x$ is

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