Advertisements
Advertisements
प्रश्न
If \[y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x, \text { then find } \left( x^2 - 1 \right) y_2 + x y_1 =\] ?
पर्याय
`-n^2y`
my
`n^2y`
None of these
Advertisements
उत्तर
\[\left( c \right) n^2 y\]
\[ y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x\]
\[\text { Differentiating the above equation with respect to x }\]
\[\left( \frac{1}{n} y^\frac{1}{n} - 1 - \frac{1}{n} y^{- \frac{1}{n} - 1} \right) y_1 = 2\]
\[\frac{1}{ny}\left( y^\frac{1}{n} - y^{- \frac{1}{n}} \right) y_1 = 2\]
\[\left( y^\frac{1}{n} - y^{- \frac{1}{n}} \right) y_1 = 2ny . . . . . \left( 1 \right)\]
\[\left( y^\frac{1}{n} - y^{- \frac{1}{n}} \right) y_2 + y_1 \left( \frac{1}{n} y^\frac{1}{n} - 1 + \frac{1}{n} y^{- \frac{1}{n} - 1} \right) y_1 = 2n y_1 \]
\[ny\left( y^\frac{1}{n} - y^{- \frac{1}{n}} \right) y_2 + {y_1}^2 \left( y^\frac{1}{n} + y^{- \frac{1}{n}} \right) = 2 n^2 y y_1 \]
\[\text{ Dividing the above equation by } y_1 \]
\[\frac{ny}{y_1}\left( y^\frac{1}{n} - y^{- \frac{1}{n}} \right) y_2 + y_1 \left( y^\frac{1}{n} + y^{- \frac{1}{n}} \right) = 2 n^2 y\]
\[\text {Putting y_1 from equation }\left( 1 \right)\]
\[\frac{\left( y^\frac{1}{n} - y^{- \frac{1}{n}} \right)^2}{2} y_2 + y_1 \left( y^\frac{1}{n} + y^{- \frac{1}{n}} \right) = 2 n^2 y . . . . . \left( 2 \right)\]
\[\text { Now,} \]
\[ \left( y^\frac{1}{n} - y^{- \frac{1}{n}} \right)^2 = \left( y^\frac{1}{n} + y^{- \frac{1}{n}} \right)^2 - 4\]
\[ \left( y^\frac{1}{n} - y^{- \frac{1}{n}} \right)^2 = 4 x^2 - 4 . . . . . \left( 3 \right)\]
\[\text { Putting the value of }\left( 3 \right)in\left( 2 \right)\]
\[\frac{4\left( x^2 - 1 \right) y_2}{2} + 2x y_1 = 2 n^2 y\]
\[\left( x^2 - 1 \right) y_2 + x y_1 = n^2 y\]
APPEARS IN
संबंधित प्रश्न
Differentiate sin (3x + 5) ?
Differentiate \[3^{x^2 + 2x}\] ?
Differentiate \[\sqrt{\frac{1 + x}{1 - x}}\] ?
Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?
Differentiate \[e^{\sin^{- 1} 2x}\] ?
Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[e^x \log \sin 2x\] ?
Differentiate \[\frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} - \sqrt{x^2 - 1}}\] ?
Differentiate \[e^{ax} \sec x \tan 2x\] ?
Differentiate \[\log \left( \cos x^2 \right)\] ?
If \[y = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] .prove that \[\frac{dy}{dx} = 1 - y^2\] ?
Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?
Find \[\frac{dy}{dx}\] in the following case \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?
If \[x \sqrt{1 + y} + y \sqrt{1 + x} = 0\] , prove that \[\left( 1 + x \right)^2 \frac{dy}{dx} + 1 = 0\] ?
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 \[y = x \sin y\] , Prove that \[\frac{dy}{dx} = \frac{\sin y}{\left( 1 - x \cos y \right)}\] ?
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 \[\left( \log x \right)^{ \log x }\] ?
Differentiate \[\left( \tan x \right)^{1/x}\] ?
Differentiate \[x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}\] ?
Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?
If \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?
If \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?
If \[y = e^{x^{e^x}} + x^{e^{e^x}} + e^{x^{x^e}}\], prove that \[\frac{dy}{dx} = e^{x^{e^x}} \cdot x^{e^x} \left\{ \frac{e^x}{x} + e^x \cdot \log x \right\}+ x^{e^{e^x}} \cdot e^{e^x} \left\{ \frac{1}{x} + e^x \cdot \log x \right\} + e^{x^{x^e}} x^{x^e} \cdot x^{e - 1} \left\{ x + e \log x \right\}\]
Find \[\frac{dy}{dx}\] , when \[x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}\] ?
If \[x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}\] ?
Differentiate x2 with respect to x3
Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with respect to \[\sec^{- 1} x\] ?
If \[y = x \left| x \right|\] , find \[\frac{dy}{dx} \text{ for } x < 0\] ?
If \[y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e\] ?
Differential coefficient of sec(tan−1 x) is ______.
If x = a(1 − cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{1}{a}\text { at } \theta = \frac{\pi}{2}\] ?
If y = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \cdot \frac{dy}{dx} + y \cos^2 x = 0\] ?
If y = cos−1 x, find \[\frac{d^2 y}{d x^2}\] in terms of y alone ?
Find the minimum value of (ax + by), where xy = c2.
