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Question
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" = ("y" + sqrt("x"^2 - "y"^2))/"x"` is
Options
`sin^-1 ("y"/"x") = 2 log |"x"| + "c"`
`sin^-1 ("y"/"x") = log |"x"| + "c"`
`sin ("y"/"x") = log |"x"| + "c"`
`sin ("y"/"x") = 2 log |"x"| + "c"`
Solution
`sin^-1 ("y"/"x") = log |"x"| + "c"`
Hint:
`"dy"/"dx" = ("y" + sqrt("x"^2 - "y"^2))/"x"`
Put y = vx ∴ `"dy"/"dx" = "v + x" "dv"/"dx"`
∴ `"v + x" "dv"/"dx" = ("vx" + sqrt("x"^2 - "v"^2"x"^2))/"x" = "v" + sqrt(1 - "v"^2)`
∴ `"x" "dv"/"dx" = sqrt(1 - "v"^2)`
∴ `int 1/sqrt(1 - "v"^2) "dv" = int 1/"x" "dx"`
∴ sin-1 v = log |x| + c
∴ sin-1 v `("y"/"x") = log |x| + "c"`.
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