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Solve the differential equation `("d"y)/("d"x) + y` = e^{−x}
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Solution
`("d"y)/("d"x) + y` = e^{−x}
The given equation is of the form
`("d"y)/("d"x) + "P"y` = Q.
where P = 1 and Q = e^{−x}
∴ I.F. = `"e"^(int^("Pd"x))`
= `"e"^(int"d"x)`
= e^{x}
∴ Solution of the given equation is
`y("I.""F.") = int"Q"("I.F.") "d"x + "c"`
∴ `y * "e"^x = int"e"^(-x) xx "e"^x "d"x + "c"`
∴ `y * "e"^x = int"e"^(-x + x) "d"x + "c"`
∴ `y * "e"^x = int"e"^0 "d"x + "c"`
∴ `y * "e"^x = int 1"d"x + "c"`
∴ ye^{x} = x + c
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