Solve the following differential equation.
` (1) e^(x) tan^(2)y dx + (e^(x) -1) sec^(2) y dy = 0`

Explanation

`e^(x) tan^(2)y dx + (e^(x) -1) sec^(2)y dy =0`
` e^(x)/(e^(x)-1) dx + (sec^(2)y)/(tan^(2)y) dy =0`
On integrating , we get ,
`int (e^(x))/(e^(x)-1) + int (sec^(2)y)/(tan^(2)y)=dy=c`
` "put " e^(x) -1=t , " " e^(x) dx = dt`
(1) becomes , ` int1/t dt+ int 1/(cos^(2)y) xx (cos^(2)y)/(sin^(2)y) dy =c `
` int 1/t dt+ int cosec^(2)y dy =c`
`log |t| + (-cot y) =c`
`log |e^(x) -1| -cot y =c`
This is the general solution