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Question
Find the derivatives of the following:
If y = etan–1x, show that (1 + x2)y” + (2x – 1)y’ = 0
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Solution
y = etan–1x
y = `"e"^(tan - 1x) (1/(1 + x^2))`
⇒ y’ = `y/(1 + x^2)`
⇒ y'(1 + x2) = y
Differentiating w.r.to x
y’ (2x) + (1 + x2)(y”) = y’
(i.e.) (1 + x2)y” + y’(2x) – y’ = 0
(i.e.) (1 + x2)y” + (2x – 1)y’ = 0
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