Given that $v=(sin u)^(1/2)$ , Show that $4v^3 (d^2v)/(du^2)+v^4+1=0$ ?

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Given that $v=(sin u)^(1/2)$ , Show that $4v^3 (d^2v)/(du^2)+v^4+1=0$ ?

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Answer 1 · 2018-01-16T16:37:23

We have:

$v = (sinu)^(1/2)$

Squaring we get:

$v^2 = sinu$ .... [A]

Implicitly differentiating wrt $u$ :

$2v \ (dv)/(du) = cos u$ .... [B]

Differentiate again wrt $u$ (using the product rule):

$(2v)( (d^2v)/(du^2)) + (2(dv)/(du))( (dv)/(du) )= -sin u$

$:. 2v (d^2v)/(du^2) + 2((dv)/(du))^2 = -sin u$

$:. 4v^3 (d^2v)/(du^2) + 4v^2((dv)/(du))^2 = -2v^2sin u$

$:. 4v^3 (d^2v)/(du^2) + (2v(dv)/(du))^2 = -2v^2sin u$

$:. 4v^3 (d^2v)/(du^2) + (cosu)^2 = -2v^2sin u \ \ \ \ \$ (from [B])

$:. 4v^3 (d^2v)/(du^2) + (1-sin^2u) = -2v^2sin u$

$:. 4v^3 (d^2v)/(du^2) + (1-v^4) = -2v^2 \ v^2$

$:. 4v^3 (d^2v)/(du^2) + 1+v^4 = 0 \ \ \ \$ QED

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Given that $v=(sin u)^(1/2)$ , Show that $4v^3 (d^2v)/(du^2)+v^4+1=0$ ?

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Given that $v=(sin u)^(1/2)$ , Show that $4v^3 (d^2v)/(du^2)+v^4+1=0$ ?