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Question
The solution of `x ("d"y)/("d"x) + y` = ex is ______.
Options
y = `"e"^x/x + "k"/x`
y = xex + cx
y = xex + k
x = `"e"^y/y + "k"/y`
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Solution
The solution of `x ("d"y)/("d"x) + y` = ex is y = `"e"^x/x + "k"/x`.
Explanation:
The given differential equation is `x ("d"y)/("d"x) + y = "e"^x`
⇒ `("d"y)/("d"x) + y/x = "e"^x/x`
Here P = `1/x` and Q = `"e"^x/x`
∴ Integrating factor I.F. = `"e"^(int 1/x "d"x)`
= `"e"^(log |x|)`
= x
So, the solution is `y xx "I"."F". = int "Q" xx "I"."F". "d"x + "k"`
⇒ `y xx x = int "e"^x/x xx x "d"x + "k"`
⇒ `y xx x = int "e"^x "d"x + "k"`
⇒ `y xx x = "e"^x + "k"`
∴ y = `"e"^x/x + "k"/x`
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