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Question
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.
Solution
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by `x"e"^(intPdx) = int "Q"_1"e"^(int P_1"d"y) "d"y + "C"`.
Explanation:
We have `("d"x)/("d"x) + "P"_1x = "Q"_1`
For solving such equation we multiply both sides by
Integrating factor = I.F. = `"e"^(int Pdx)`
So we get `"e"^(intPdx) (("d"x)/("d"y) + "P"_1x) = "Q"_1"e"^(intPdx)`
⇒ `("d"x)/("d"y) "e"^(intPdx) + "P"_1"e"^(intPdy) = "Q"_1"e"^(intP_1dy)`
⇒ `"d"/("d"y)(x"e"^(intP_1dy)) = "Q"_1"e"^(intP_1dy)`
⇒ `int "d"/("d"y) (x"e"^(intP_1dy))"d"y = int "Q"_1"e"^(intP_1dy) "d"y`
⇒ `x"e"^(intP_1"d"y) = int"Q"_1"e"^(intP_1dy) "d"y + "C"`
This is the required solution of the given differential equation.
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