The general solution of frac{dy}{dx} = x + si | vedclass

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(B) Given the differential equation: $\frac{dy}{dx} = x + \sin x \cos y + x \cos y + \sin x$ Factor the right side: $\frac{dy}{dx} = x(1 + \cos y) + \

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$	an \frac{y}{2} = \frac{x^2}{2} - \cos x + C$

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(B) Given the differential equation: $\frac{dy}{dx} = x + \sin x \cos y + x \cos y + \sin x$ Factor the right side: $\frac{dy}{dx} = x(1 + \cos y) + \sin x(1 + \cos y)$ $\frac{dy}{dx} = (x + \sin x)(1 + \cos y)$ Separate the variables: $\frac{dy}{1 + \cos y} = (x + \sin x) dx$ Using the identity $1 + \cos y = 2 \cos^2 \frac{y}{2}$, we get: $\frac{dy}{2 \cos^2 \frac{y}{2}} = (x + \sin x) dx$ $\frac{1}{2} \sec^2 \frac{y}{2} dy = (x + \sin x) dx$ Integrating both sides: $\int \frac{1}{2} \sec^2 \frac{y}{2} dy = \int (x + \sin x) dx$ $	an \frac{y}{2} = \frac{x^2}{2} - \cos x + C$

The general solution of $\frac{dy}{dx} = x + \sin x \cos y + x \cos y + \sin x$ is

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