1.

solution of differential equation `xcosx(dy)/(dx)+y(xsinx+cosx)=1` isA. `xy=sin x+C cos x`B. `xy+cosx+C sin x = 0`C. ` xy +sec x+C sin x=0`D. None of the above (where , C is arbitrary constant)

Answer» Correct Answer - a
Given , `x cos x ((dy)/(dx)) + y ( x sin x + cos x ) = 1`
We can write ,
` (dy)/(dx) +y (tanx.+1/x)=(1/(xcosx))`
which is linear differential equation ltbRgt Here, ` P = tan x + 1/x , Q = 1/(x cos x)`
` :. IF = e^((tan x + 1/x)dx) = e ^(log sec x + log x ) = e^(log ( x sec x))= x sec x `
Now, required solution is
`y (IF) = int IF * Q dx +C`
` rArr y ( x sec x) = int x sec x * 1/(x cos x) dx +C`
` rArr xy sec x = int sec^(2) x dx+C`
` rArr xy sec x = tan x +C`
On multiplying both sides by cos x , we get
`xy = sin x + C cos x`


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