Advertisements
Advertisements
प्रश्न
If y = a xn + 1 + bx−n and \[x^2 \frac{d^2 y}{d x^2} = \lambda y\] then write the value of λ ?
Advertisements
उत्तर
\[y = a x^{n + 1} + b x^{- n} \]
\[\text { and }x^2 \frac{d^2 y}{d x^2} = \lambda y\]
\[\text { Now }, \]
\[\frac{d y}{d x} = a\left( n + 1 \right) x^n - bn x^{- n - 1} \]
\[\text{ and } \frac{d^2 y}{d x^2} = an\left( n + 1 \right) x^{n - 1} - bn\left( - n - 1 \right) x^{- n - 2} \]
\[\text { Now,} x^2 \frac{d^2 y}{d x^2} = \lambda y \left[ \text { Given } \right]\]
\[ \Rightarrow x^2 \left[ an\left( n + 1 \right) x^{n - 1} + bn\left( n + 1 \right) x^{- n - 2} \right] = \lambda\left( a x^{n + 1} + b x^{- n} \right)\]
\[ \Rightarrow an\left( n + 1 \right) x^{n + 1} + bn\left( n + 1 \right) x^{- n} = \lambda\left( a x^{n + 1} + b x^{- n} \right)\]
\[ \Rightarrow n\left( n + 1 \right)\left( a x^{n + 1} + b x^{- n} \right) = \lambda\left( a x^{n + 1} + b x^{- n} \right)\]
\[ \Rightarrow \lambda = n\left( n + 1 \right)\]
APPEARS IN
संबंधित प्रश्न
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 tan (x° + 45°) ?
Differentiate \[e^{\sin} \sqrt{x}\] ?
Differentiate \[\sqrt{\frac{a^2 - x^2}{a^2 + x^2}}\] ?
Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?
Differentiate \[\log \sqrt{\frac{1 - \cos x}{1 + \cos x}}\] ?
Differentiate \[\sqrt{\tan^{- 1} \left( \frac{x}{2} \right)}\] ?
Differentiate the following with respect to x:
\[\cos^{- 1} \left( \sin x \right)\]
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 \[x^{2/3} + y^{2/3} = a^{2/3}\] ?
Find \[\frac{dy}{dx}\] in the following case \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?
If \[x y^2 = 1,\] prove that \[2\frac{dy}{dx} + y^3 = 0\] ?
If \[\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find} \frac{dy}{dx}\] ?
Differentiate \[e^{x \log x}\] ?
Differentiate \[x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}\] ?
If \[x^x + y^x = 1\], prove that \[\frac{dy}{dx} = - \left\{ \frac{x^x \left( 1 + \log x \right) + y^x \cdot \log y}{x \cdot y^\left( x - 1 \right)} \right\}\] ?
Find the derivative of the function f (x) given by \[f\left( x \right) = \left( 1 + x \right) \left( 1 + x^2 \right) \left( 1 + x^4 \right) \left( 1 + x^8 \right)\] and hence find `f' (1)` ?
If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] , when \[x = b \sin^2 \theta \text{ and } y = a \cos^2 \theta\] ?
Find \[\frac{dy}{dx}\] , when \[x = \frac{3 at}{1 + t^2}, \text{ and } y = \frac{3 a t^2}{1 + t^2}\] ?
If \[x = a\left( t + \frac{1}{t} \right) \text{ and y } = a\left( t - \frac{1}{t} \right)\] ,prove that \[\frac{dy}{dx} = \frac{x}{y}\]?
\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?
Let g (x) be the inverse of an invertible function f (x) which is derivable at x = 3. If f (3) = 9 and `f' (3) = 9`, write the value of `g' (9)`.
If \[y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\] ,Then, write the value of \[\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \] ?
If \[y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e\] ?
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text{ find } \frac{dy}{dx}\] ?
If f (x) is an even function, then write whether `f' (x)` is even or odd ?
If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] _____________ .
If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
If \[\sin^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \text { log a then } \frac{dy}{dx}\] is equal to _____________ .
If y = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
If \[y = e^{a \cos^{- 1}} x\] ,prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - a^2 y = 0\] ?
\[\text { If x } = \cos t + \log \tan\frac{t}{2}, y = \sin t, \text { then find the value of } \frac{d^2 y}{d t^2} \text { and } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
\[\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3} \] ?
f(x) = 3x2 + 6x + 8, x ∈ R
