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Question

Solve (1+y)tan2x+tanxdydx+y=0


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Solution

Simplify and Integrate the equation:

Given that,

(1+y)tan2x+tanxdydx+y=0tanxdydx+tan2x+1y=-tan2xTakingyascommontermtanxtanxdydx+tan2xtanx+1tanxy=-tan2xtanxDividngtheequationbytanxdydx+tanx+cotxy=-tanx1tanx=cotx

Integrating the equation

I.F.=etanx+cotxdx=elnsecx+lnsinx=elnsecx·sinx=tanx

Now,

ytanx=-tan2xdxytanx=-(sec2x-1)dxtan2x=sec2x-1ytanx=-tanx+x+c

Therefore, the solution of (1+y)tan2x+tanxdydx+y=0 is -tanx+x+c.


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