Solve the following differential equation: cosecxlogydydx+x²y²=0

cosecx log ydydx+x²y²=0⇒cosecx log ydydx=−x²y²Separating the variables, we get⇒log(y)y²dy=−x²cosecxdx⇒∫(logy)(1/y²)dy=−∫x²sinxdx⇒logy(-1/y)-∫(-1/y²)dy=x²cosx-∫2xcosxdx+C⇒-logy/y+1/y=x²cosx-[2xsinx-∫2sinxdx]+C⇒-logy/y+1/y=x²cosx-[2xsinx+2cosx]+Clogy/y+1/y+x²cosx-[2xsinx+2cosx]+C=0