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Question
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
Options
True
False
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Solution
This statement is True.
Explanation:
The given differential equation is `x("d"y)/("d"x) = y + x tan (y/x)`
`x ("d"y)/("d"x) = -x tan (y/x)` = y
⇒ `("d"y)/("d"x) - tan (y/x) = y/x`
⇒ `("d"y)/("d"x) = y/x + tan (y/x)`
Put y = vx
⇒ `("d"y)/("d"x) = "v" + x "dv"/"dx"`
⇒ `"v" + x * "dv"/"dx" = "vx"/x + tan ("vx"/x)`
⇒ `"v" + x "dv"/"dx" = "v" + tan "v"`
⇒ `x "dv"/"dx" = tan "v"`
⇒ `"dv"/tan"v" = ("d"x)/x`
⇒ `cot "v" "dv" = ("d"x)/x`
Integrating both sides, we get
`int cot "v" "dv" = int ("d"x)/x`
⇒ `log sin "v" = log x + log "c"`
⇒ `log sin "v" - log x = log "c"`
⇒ `log sin y/x = log x"c"`
∴ `sin y/x` = xc
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