Solve the following differential equation: ` cosec x log y(dy)/(dx)+\ x^2y^2=0`

Explanation

` {cosecx} log y frac{d y}{d x}+x^{2} y^{2}=0`


`Rightarrow {cosecx} log y frac{d y}{d x}=-x^{2} y^{2}`


Separate the variables


`Rightarrow frac{log (y)}{y^{2}} d y=frac{-x^{2}}{cosecx} d x `


`Rightarrow int(log y)(frac{1}{y^{2}} d y)=-int x^{2} sin x d x `


`Rightarrow log y(-frac{1}{y})-int(-frac{1}{y^{2}}) d y=x^{2} cos x-int 2 x cosxdx+c`


`Rightarrow-frac{log y}{y}+int y^{-2} d y=x^{2} cos x-2[x sin x-int 1 . sinxdx]+c`


`Rightarrow-frac{log y}{y}-y^{-1}=x^{2} cos x-2 x sin x-2 cos x+C `


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