Solve the following differential equation yxdydx = x2 + 2y2 - Mathematics and Statistics

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Sum

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 = c2x 

  Is there an error in this question or solution?
Chapter 1.8: Differential Equation and Applications - Q.5

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