The solution of the differential equation {x^ | vedclass

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(A) Given differential equation: ${x^4}\frac{{dy}}{{dx}} + {x^3}y + 	ext{cosec}(xy) = 0$Multiply by $dx$: ${x^4}dy + {x^3}y\,dx + 	ext{cosec}(xy)\,d

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$2\cos(xy) + x^{-2} = c$

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(A) Given differential equation: ${x^4}\frac{{dy}}{{dx}} + {x^3}y + 	ext{cosec}(xy) = 0$Multiply by $dx$: ${x^4}dy + {x^3}y\,dx + 	ext{cosec}(xy)\,dx = 0$Rearrange the terms: ${x^3}(x\,dy + y\,dx) + 	ext{cosec}(xy)\,dx = 0$Recognizing the differential of a product: ${x^3}d(xy) + 	ext{cosec}(xy)\,dx = 0$Separate the variables: $\frac{d(xy)}{	ext{cosec}(xy)} + \frac{dx}{x^3} = 0$This simplifies to: $\sin(xy)\,d(xy) + x^{-3}\,dx = 0$Integrating both sides: $\int \sin(xy)\,d(xy) + \int x^{-3}\,dx = \int 0$Result of integration: $-\cos(xy) + \frac{x^{-2}}{-2} = C_1$Multiply by $-2$: $2\cos(xy) + x^{-2} = c$ (where $c = -2C_1$).

The solution of the differential equation ${x^4}\frac{{dy}}{{dx}} + {x^3}y + \text{cosec}(xy) = 0$ is equal to

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