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Question
Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
`y = sqrt(a^2 - x^2 ) x in (-a,a) : x + y dy/dx = 0(y != 0)`
Solution
We have, `y = sqrt (a^2 - x^2)` ...(i)
Differentiating (I) w.r.t. x, we get
`y' = (1xx (-2x))/(2sqrt(a^2 - x^2))`
⇒ `y' = (-x)/sqrt (a^2 - x^2)`
⇒ `y' = (-x)/y` (Using (i))
⇒ yy' = -x
⇒ x + yy' = 0
∴ `y = sqrt (a^2 - x^2)` is a solution of the given differential equation.
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