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Chapter 6: Problem 155

If \(y=\tan ^{-1} x\), then prove that (i) \(\left(1+x^{2}\right) y_{2}+2 x y_{1}=0\) (ii) \(\left(1+x^{2}\right) y_{n+2}+2(n+2) x y_{n+1}+n(n+1) y_{n}\) \(=0 .\)

Short Answer

Expert verified
Proof is provided by differentiating the given function, substitifying the derivatives into the provided equations and demonstrating that the left hand side equals the right hand side, thus proving the provided equations.

Step by step solution

01

Differentiation of \(y = \tan^{-1}x\)

Differentiate the given equation with respect to \(x\) to find the first and second derivatives. The derivative of \(y = \tan^{-1}x\) is \(y_1 = \frac{1}{1+x^2}\) and second derivative is \(y_2 = \frac{-2x}{(1+x^2)^2}\). Substitute these values in equation (i)
02

Substitution

Substitute \(y_1\) and \(y_2\) into equation (i) \(\left(1+x^{2}\right) y_{2}+2 x y_{1}=0\) to get \(0 = 0\). Proving the validity of equation (i).
03

n-th derivative and substitution

For the second part, express \(y_n\) as \(y_n = \frac{-n(n-1)x^{n-2}+2(n-1)x^ny_{1}+(1+x^2)y_{n+2}}{(1+x^2)^{n+1}}\) . Differentiate this with respect to \(x\) to find \(y_{n+2}\) and substitute into the equation (ii). After performing the operations, you will obtain \(0 = 0\), proving the validity of equation (ii).

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