The solution of the differential equation dy | vedclass

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Question

(A) Given the differential equation: $dy - \sin x \sin y \, dx = 0$. Rearranging the terms to separate the variables: $dy = \sin x \sin y \, dx$. Divi

Vedclass · Accepted Answer

$e^{\cos x} 	an \frac{y}{2} = c$

Vedclass · Answer

(A) Given the differential equation: $dy - \sin x \sin y \, dx = 0$. Rearranging the terms to separate the variables: $dy = \sin x \sin y \, dx$. Dividing both sides by $\sin y$: $\frac{dy}{\sin y} = \sin x \, dx$. Integrating both sides: $\int \csc y \, dy = \int \sin x \, dx$. Using the standard integral $\int \csc y \, dy = \ln |	an \frac{y}{2}|$: $\ln |	an \frac{y}{2}| = -\cos x + C$. Taking the exponential of both sides: $	an \frac{y}{2} = e^{-\cos x + C} = e^{-\cos x} \cdot e^C$. Let $e^C = c_1$,then $	an \frac{y}{2} = c_1 e^{-\cos x}$. Multiplying by $e^{\cos x}$: $e^{\cos x} 	an \frac{y}{2} = c_1$.

The solution of the differential equation $dy - \sin x \sin y \, dx = 0$ is

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