Advertisements
Advertisements
Question
\[\text { If y } = a \left\{ x + \sqrt{x^2 + 1} \right\}^n + b \left\{ x - \sqrt{x^2 + 1} \right\}^{- n} , \text { prove that }\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0 \]
Disclaimer: There is a misprint in the question,
\[\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0\] must be written instead of
\[\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0 \] ?
Advertisements
Solution
\[\text { We have,} \]
\[y = a \left\{ x + \sqrt{x^2 + 1} \right\}^n + b \left\{ x - \sqrt{x^2 + 1} \right\}^{- n} . . . (1)\]
\[\text { Differentiating y with respect to x, we get }\]
\[\frac{d y}{d x} =\text { an} \left\{ x + \sqrt{x^2 + 1} \right\}^{n - 1} \left( 1 + \frac{1}{2\sqrt{x^2 + 1}} \times 2x \right) - bn \left\{ x - \sqrt{x^2 + 1} \right\}^{- n - 1} \left( 1 - \frac{1}{2\sqrt{x^2 + 1}} \times 2x \right)\]
\[ = \text { an }\left\{ x + \sqrt{x^2 + 1} \right\}^{n - 1} \left( 1 + \frac{x}{\sqrt{x^2 + 1}} \right) - bn \left\{ x - \sqrt{x^2 + 1} \right\}^{- n - 1} \left( 1 - \frac{x}{\sqrt{x^2 + 1}} \right)\]
\[ = \text { an }\left\{ x + \sqrt{x^2 + 1} \right\}^{n - 1} \left( \frac{\sqrt{x^2 + 1} + x}{\sqrt{x^2 + 1}} \right) - bn \left\{ x - \sqrt{x^2 + 1} \right\}^{- n - 1} \left( \frac{\sqrt{x^2 + 1} - x}{\sqrt{x^2 + 1}} \right)\]
\[ = \text { an } \left\{ x + \sqrt{x^2 + 1} \right\}^{n - 1} \left( \frac{x + \sqrt{x^2 + 1}}{\sqrt{x^2 + 1}} \right) + bn \left\{ x - \sqrt{x^2 + 1} \right\}^{- n - 1} \left( \frac{x - \sqrt{x^2 + 1}}{\sqrt{x^2 + 1}} \right)\]
\[ = \left\{ a \left\{ x + \sqrt{x^2 + 1} \right\}^n \left( \frac{n}{\sqrt{x^2 + 1}} \right) + b \left\{ x - \sqrt{x^2 + 1} \right\}^{- n} \right\}\left( \frac{n}{\sqrt{x^2 + 1}} \right)\]
\[ = \left( \frac{n}{\sqrt{x^2 + 1}} \right)y \left[ \text { From }(1) \right]\]
\[ \Rightarrow \sqrt{x^2 + 1}\frac{d y}{d x} = ny\]
\[\text { Squaring both sides, we get }\]
\[\left( x^2 + 1 \right) \left( \frac{d y}{d x} \right)^2 = n^2 y^2 . . . (2)\]
\[\text{ Differentiating (2) with respect to x, we get }\]
\[\left( x^2 + 1 \right)2\frac{d y}{d x} \times \frac{d^2 y}{d x^2} + 2x \left( \frac{d y}{d x} \right)^2 = n^2 \left( 2y\frac{d y}{d x} \right)\]
\[ \Rightarrow \left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\left( \frac{d y}{d x} \right) = n^2 \left( y \right)\]
\[ \Rightarrow \left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\left( \frac{d y}{d x} \right) - n^2 y = 0\]
\[\text { Hence, }\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0 .\]
APPEARS IN
RELATED QUESTIONS
If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 60º.
Differentiate the following functions from first principles \[e^\sqrt{2x}\].
Differentiate the following functions from first principles log cosec x ?
Differentiate \[e^{\sin} \sqrt{x}\] ?
Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?
Differentiate \[\frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} - \sqrt{x^2 - 1}}\] ?
Prove that \[\frac{d}{dx} \left\{ \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{- 1} \frac{x}{a} \right\} = \sqrt{a^2 - x^2}\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{1 + \sqrt{1 - x^2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{4x}{1 - 4 x^2} \right), - \frac{1}{2} < x < \frac{1}{2}\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
If \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, - \frac{1}{2} < x < 0, \text{ find } \frac{dy}{dx} \] ?
Find \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?
If \[\sin^2 y + \cos xy = k,\] find \[\frac{dy}{dx}\] at \[x = 1 , \] \[y = \frac{\pi}{4} .\]
If \[y = \left\{ \log_{\cos x} \sin x \right\} \left\{ \log_{\sin x} \cos x \right\}^{- 1} + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right), \text{ find } \frac{dy}{dx} \text{ at }x = \frac{\pi}{4}\] ?
Differentiate \[x^{1/x}\] with respect to x.
Find \[\frac{dy}{dx}\] \[y = \left( \sin x \right)^{\cos x} + \left( \cos x \right)^{\sin x}\] ?
If \[\left( \sin x \right)^y = \left( \cos y \right)^x ,\], prove that \[\frac{dy}{dx} = \frac{\log \cos y - y cot x}{\log \sin x + x \tan y}\] ?
If \[e^{x + y} - x = 0\] ,prove that \[\frac{dy}{dx} = \frac{1 - x}{x}\] ?
If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
If \[xy \log \left( x + y \right) = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
If \[y = x \sin y\] , prove that \[\frac{dy}{dx} = \frac{y}{x \left( 1 - x \cos y \right)}\] ?
If \[y = \left( \tan x \right)^{\left( \tan x \right)^{\left( \tan x \right)^{. . . \infty}}}\], prove that \[\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}\] ?
If \[\left| x \right| < 1 \text{ and y} = 1 + x + x^2 + . . \] to ∞, then find the value of \[\frac{dy}{dx}\] ?
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
If f (x) = logx2 (log x), the `f' (x)` at x = e is ____________ .
If \[y = \left( 1 + \frac{1}{x} \right)^x , \text{then} \frac{dy}{dx} =\] ____________.
If \[x = a \cos^3 \theta, y = a \sin^3 \theta, \text { then } \sqrt{1 + \left( \frac{dy}{dx} \right)^2} =\] ____________ .
If \[y = \log \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] __________ .
If x = a (1 − cos3θ), y = a sin3θ, prove that \[\frac{d^2 y}{d x^2} = \frac{32}{27a} \text { at } \theta = \frac{\pi}{6}\]?
If x = sin t, y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?
If y = 500 e7x + 600 e−7x, show that \[\frac{d^2 y}{d x^2} = 49y\] ?
If y = sin (log x), prove that \[x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\] ?
If \[f\left( x \right) = \frac{\sin^{- 1} x}{\sqrt{1 - x^2}}\] then (1 − x)2 f '' (x) − xf(x) =
If \[y = \tan^{- 1} \left\{ \frac{\log_e \left( e/ x^2 \right)}{\log_e \left( e x^2 \right)} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)\], then \[\frac{d^2 y}{d x^2} =\]
If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x3 y2 =
Differentiate `log [x+2+sqrt(x^2+4x+1)]`
Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume, and radius r.
