The general solution of frac{dy}{dx} = cos^2( | vedclass

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(A) Given the differential equation: $\frac{dy}{dx} = \cos^2(x-y-1)$ $(i)$ Let $x-y-1 = p$. Differentiating with respect to $x$,we get $1 - \frac{dy}{

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$C - \cot(x-y-1)$

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(A) Given the differential equation: $\frac{dy}{dx} = \cos^2(x-y-1)$ $(i)$ Let $x-y-1 = p$. Differentiating with respect to $x$,we get $1 - \frac{dy}{dx} = \frac{dp}{dx}$,which implies $\frac{dy}{dx} = 1 - \frac{dp}{dx}$. Substituting this into equation $(i)$: $1 - \frac{dp}{dx} = \cos^2 p$ $\frac{dp}{dx} = 1 - \cos^2 p = \sin^2 p$ $\frac{dp}{\sin^2 p} = dx$ $\operatorname{cosec}^2 p \, dp = dx$ Integrating both sides: $\int \operatorname{cosec}^2 p \, dp = \int dx$ $-\cot p = x + C'$ $x = -C' - \cot p$ Let $C = -C'$,then $x = C - \cot(x-y-1)$.

The general solution of $\frac{dy}{dx} = \cos^2(x-y-1)$ is given by $x=$

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