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Question
Solve the following differential equation
`yx ("d"y)/("d"x)` = x2 + 2y2
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Solution
`yx ("d"y)/("d"x)` = x2 + 2y2
∴ `("d"y)/("d"x) = (x^2 + 2y^2)/(xy)` ......(i)
Put y = vx ......(ii)
Differentiating w.r.t. x, we get
`("d"y)/("d"x) = "v" + x "dv"/("d"x)` ......(iii)
Substituting (ii) and (iii) in (i), we get
`"v" + x "dv"/("d"x) = (x^2 + 2"v"^2x^2)/(x("v"x))`
∴ `"v" + x "dv"/("d"x) = (x^2(1 + 2"v"^2))/(x^2"v")`
∴ `x "dv"/("d"x) = (1 + 2"v"^2)/"v" - "v"`
= `(1 + "v"^2)/"v"`
∴ `"v"/(1 + "v"^2) "dv" = 1/x "d"x`
Integrating on both sides, we get
`1/ int (2"v")/(1 +"v"^2) "dv" = int "dv"/x`
∴ `1/2 log|1 + "v"^2|` = log |x| + log |c|
∴ log |1 + c2| = 2 og |x| + 2log |c|
= log |x2| + log |c2|
∴ log |1 + v2| = log |c2x2|
∴ 1 + v2 = c2x2
∴ `1 + y^2/x^2` = c2x2
∴ x2 + y2 = c2x4
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