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Question
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
Options
xy = – ex
xy = – e-x
xy = – 1
y = 2ex – 1
Solution
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by y = 2ex – 1.
Explanation:
The given differential equation is `("d"y)/("d"x) - y` = 1
Here, P = –1, Q = 1
∴ Integrating factor, I.F. = `"e"^(intPdx)`
= `"e"^(int -1"d"x)`
= `"e"^-x`
So, the solution is `y xx "I"."F". = int "Q" ."I"."F". "d"x + "c"`
⇒ `y xx "e"^-x = int 1."e"^-x "d"x + "c"`
⇒ `y * "e"^-x = -"e"^-x + "c"`
Put x = 0, y = 1
⇒ `1. "e"^0 = - "e"^0 + "c"`
⇒ 1 = `-1 + "c"`
∴ c = 2
So the equation is `y * "e"^-x = -"e"^-x + 2`
⇒ y = `-1 + 2"e"^x`
= `2"e"^x - 1`.
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