1-If-y-x-n-1-log-x-then-prove-that-x-2-d-2-y-dx-2-3-2n-x-dy-dx-n-1-2-y-0-2-If-mtan-cos-2-ntan-cos-2-then-prove-that-1-2-tan-1-n-m-n-m-tan-

Question Number 65601 by AnjanDey last updated on 31/Jul/19

1 . If y = x^{n - 1} \log x, then prove that, x^{2} \frac{d^{2} y}{{d x}^{2}} + (3 - 2 n) x \frac{d y}{d x} + {(n - 1)}^{2} y = 0

2 . I f \frac{mtan (α - θ)}{\cos^{2} θ} = \frac{n \tan θ}{\cos^{2} (α - θ)}, then prove that, θ = \frac{1}{2} [α - \tan^{- 1} (\frac{n - m}{n + m} \tan α)]

Commented by Prithwish sen last updated on 01/Aug/19

2 . \frac{\sin (α - θ) \cos (α - θ)}{\sin θ \cos θ} = \frac{n}{m} \Rightarrow \frac{\sin 2 (α - θ)}{\sin 2 θ} = \frac{n}{m}

\Rightarrow \frac{\sin 2 (α - θ) - \sin 2 θ}{\sin 2 (α - θ) + \sin 2 θ} = \frac{n - m}{n + m}

\Rightarrow \frac{\tan (α - 2 θ)}{\tan α} = \frac{n - m}{n + m}

1 . \frac{dy}{dx} = (n - 1) x^{n - 2} logx + x^{n - 2}

\frac{d^{2} y}{{dx}^{2}} = (n - 1) (n - 2) x^{n - 3} logx + (n - 1) x^{n - 3} + (n - 2) x^{n - 3}

x^{2} \frac{d^{2} y}{{dx}^{2}} = (n - 1) (n - 2) x^{n - 1} logx + (n - 1 + n - 2) x^{n - 1}

= - {(n - 1)}^{2} x^{n - 1} logx + (n - 1) (2 n - 3) x^{n - 1} logx + (2 n - 3) x^{n - 1}

= - {(n - 1)}^{2} y - (3 - 2 n) x [(n - 1) x^{n - 2} logx + x^{n - 2}]

\Rightarrow x^{2} \frac{d^{2} y}{{dx}^{2}} + (3 - 2 n) \frac{dy}{dx} + {(n - 1)}^{2} y = 0

please check .