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Question
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
Options
True
False
Solution
This statement is True.
Explanation:
The given differential equation is `("d"y)/("d"x) = (x + 2y)/x`
⇒ `("d"y)/("d"x) = 1 + 2 y/x`
⇒ `("d"y)/("d"x) = (2y)/x` = 1
Here, P = `(-2)/x` and Q = 1
Integrating factor I.F. = `"e"^(int(-2)/x "d"x)`
= `"e"^(-2 log x)`
= `"e"^(log x^-2)`
= `1/x^2`
∴ Solution is `y xx "I"."F". = int "Q" xx "I"."F". "d"x + "c"`
⇒ `y xx 1/x^2 = int 1 xx 1/x^2 "d"x + "c"`
⇒ `y/x^2 = int 1/x^2 "d"x + "c"`
⇒ `y/x^2 = - 1/x + "c"`
⇒ y = `-x + "c"x^2`
⇒ y + x = cx2
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