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Question
Solve: ydx – xdy = x2ydx.
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Solution
Given equation is ydx – xdy = x2ydx.
⇒ ydx – x2y dx = xdy
⇒ y(1 – x2)dx = xdy
⇒ `((1 - x^2)/x)"d"x = "dy"/y`
⇒ `(1/x - x)"d"x = "dy"/y`
Integrating both sides we get
`int(1/x - x)"d"x = int "dy"/y`
⇒ `log x - x^2/2` = log y + log c
⇒ `log x - x^2/2` = log yc
⇒ log y – log c = `x^2/2`
⇒ `log x/(y"c") = x^2/2`
⇒ `x/(y"c") = "e"^(x^2/2)`
⇒ `(y"c")/x = "e"^((-x^2)/2`
⇒ yc = `x"e"^((-x^2)/2`
∴ y = `1/"c" * x"e"^((-x^2)/2`
⇒ y = `"k"x"e"^((-x^2)/2` ......`[because "k" = 1/"c"]`
Hence, the required solution is y = `"k"x"e"^((-x^2)/2`.
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