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Question
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solution
Given equation is `"dy"/"dx"` = e–2y
⇒ `"dy"/"e"^(-2y)` = dx
⇒ `"e"^(2y) * "d"y` = dx
Integrating both sides, we get
`int "e"^(2y) "d"y = int "d"x`
⇒ `1/2 "e"^(2y)` = x + c
Put y = 0 and x = 5
⇒ `1/2 "e"^0` = 5 + c
⇒ c = `1/2 - 5 = - 9/2`
Now putting y = 3, we get
`1/2 "e"^6 = x - 9/2`
⇒ x = `1/2 "e"^6 + 9/2`
Hence the required value of x =`1/2 ("e"^6 + 9)`.
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