Solve the following differential equation: yxdydxxydydxy2-x2dydx=xydydx - Mathematics and Statistics

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Sum

Solve the following differential equation:

`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`

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Solution

`"y"^2 - "x"^2 "dy"/"dx" = "xy""dy"/"dx"`

∴ `"x"^2 "dy"/"dx" + "xy""dy"/"dx" = "y"^2`

∴ `("x"^2 + "xy")"dy"/"dx" = "y"^2`

∴ `"dy"/"dx" = "y"^2/("x"^2 + "xy")`      ......(1)

Put y = vx

∴ `"dy"/"dx" = "v + x" "dv"/"dx"`

∴ (1) becomes, `"v + x""dv"/"dx" = ("v"^2"x"^2)/("x"^2 + "x"*"vx") = "v"^2/(1 + "v")`

∴ `"x""dv"/"dx" = "v"^2/(1 + "v") - "v" = ("v"^2 - "v" - "v"^2)/(1 + "v")`

∴ `"x""dv"/"dx" = (- "v")/(1 + "v")`

∴ `(1 + "v")/"v" "dv" = - 1/"x" "dx"`

Integrating, we get

`int (1 + "v")/"v" "dv" = - int 1/"x" "dx"`

`int (1/"v" + 1)"dv" = - int1/"x" "dx"`

∴ `int 1/"v" "dv" + int 1 "dv" = - int 1/"x" "dx"`

∴ log |v| + v = - log |x| + c

∴ log `|"y"/"x"| + "y"/"x" = - log |"x"| + "c"`

∴ log |y| - log |x| + `"y"/"x"` = - log |x| + c

∴ `"y"/"x" + log |"y"| = "c"`

This is the general solution.

  Is there an error in this question or solution?
Chapter 6: Differential Equations - Exercise 6.4 [Page 203]

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