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3e^(x) tan y dx + (1+e^(x)) sec^(2) dy =...

`3e^(x) tan y dx + (1+e^(x)) sec^(2) dy =0 , " when" x = 0 and y = pi`

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`3e^(x) tan y dx + (1+e^(x)) sec^(2) y dy =0`
`(3e^(x))/(1 +e^(x)) dx + (sec^(2)y)/(tan y) dy=0`
Intergrating , we set
` 3 int (e^(x))/(1 +e^(x)) dx + int (sec^(2)y)/(tan y) dy =c_(1)`
each of these integrals of the type
` int (f'(x))/(f(x)) dx = log |f (x)|+c`
the general solution is
` 3 log |1 +e^(x)| + log |tany | = log c, " when " c_(1) = log c`
`log | (1+e^(x))^(3)| + log | tan y| = log c`
` log | (1+e^(x))^(3) tan y| = log c `
Now , x = 0 and ` y =pi`
` (1+e^(0))^(3) tan pi =c " " therefore c=0`
the particular solution is
` (1 + e^(x))^(3) tan y = 0`
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