solution of differential equation `xcosx(dy)/(dx)+y(xsinx+cosx)=1` is

Explanation

It is Given `x \cos x(frac{d y }{ d x})+y(x \sin x+\cos x)=1`

`frac{d y}{d x}+\frac{y(x \sin x+\cos x)}{x \cos x}=\frac{1}{x \cos x} `

`frac{d y}{d x}+(\frac{x \sin x}{x \cos x}+\frac{\cos x}{x \cos x})^{y}=\frac{1}{x \cos x} `

`frac{d y}{d x}+(\tan x+\frac{1}{x}) y=\frac{1}{x \cos x}`

It is linear differential equation in the form `(frac{d y }{ d x})+P y=Q`

where `P=\tan x+(1 / x)` and `(\sec x / x)`

`therefore(I . F .)=e^{\int(\tan x+(1 / x)) d x} `

`=e^{\int \tan x+j d x / x}=e^{\log \sec x+\log x}=e^{\log (x \sec x)}`

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