Question

If y = (tan–1 x)2 , show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2

Answer 1

Given :    y = ( Tan⁻¹x)²

To find : Show that (x²  + 1)²y₂  + 2x(x² + 1) y₁   = 2  

Solution:

y₁ = y' = dy/dx    & y₂  = y''   = d²y/dx²

y = ( Tan⁻¹x)²

Differentiating wrt x

=> y₁  = 2Tan⁻¹x . (1/(1 + x²)

=> ( 1 + x²)y₁  = 2Tan⁻¹x  

Differentiating wrt x again

=>  ( 1 + x²)y₂  + 2xy₁   = 2/(1 + x²)

=> ( 1 + x²)²y₂  + 2x(1 + x²)y₁   = 2

=> (x²  + 1)²y₂  + 2x(x² + 1) y₁   = 2

QED

Proved

(x²  + 1)²y₂  + 2x(x² + 1) y₁   = 2  

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