Find the particular solution of the differential equation `(dx)/(dy)+xcoty=2y + y^2cot y ,\ y!=0` given that `x=0\ w h e n\ y=pi/2` .

Explanation

We have `frac{d x}{d y}+x {coty}=2 y+y^{2} {coty},(y equiv 0)`


It is clear that this ia linear differential equation of the form `frac{d x}{d y}+P(y) x=Q(y)`


Here, `{P}({y})={coty}, {Q}({y})=2 {y}+{y}^{2} {coty}`


Integrating factor `=e^{int text { cotydy }}=e^{{logsiny}}=sin y`


So, solution is given by, `x sin y=int sin y(2 y+y^{2} {coty}) d y`


`x sin y=int 2 y sin y d y+int y^{2} cos y d y`


`xsiny =int 2 y sin y d y+y^{2} int cos y d y-int(frac{d(y^{2})}{d y} int cos y d y) d y`


`x sin y=int 2 y sin y d y+y^{2} sin y-int 2 y sin y d y`


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