Find the general solution of each of the following differential equations:
`e^(x) tan y dx +(1-e^(x))sec^(2)y dy =0`

Options

A. `siny =c (1-e^(x))`
B. ` cos y =c (1-e^(x))`
C. ` cot y =c (1 -e^(x))`
D. none of the above

Answer

C

Explanation

Given diff equation can be written as
` (e^(x))/(1-e^(x)) dx = -(sec^(2)y)/(tan y) dy`
On integrating both the sides, we get
` inf e^(x)/(1-e^(x)) dx = - int (sec^(2) y)/(tan y) dy`
` Rightarrow -log (1-e^(x)) = - log ( tan y) + log c`
` Rightarrow log (tan y) = log c + log (1 -e^(x))`
` Rightarrow log (tan y) = log [c (1-e^(x)]`
` Rightarrow tan y = c (1-e^(x))`
where, 'c' is the constant of intergration