Advertisements
Advertisements
प्रश्न
If y = (sin−1 x)2, prove that (1 − x2)
\[\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?
Advertisements
उत्तर
Here,
\[y = \left( \sin^{- 1} x \right)^2 \]
\[\text { Now,} \]
\[ y_1 = 2 \sin^{- 1} x \frac{1}{\sqrt{1 - x^2}}\]
\[ \Rightarrow y_2 = \frac{2}{1 - x^2} + \frac{2x \sin^{- 1} x}{\left( 1 - x^2 \right)^{3/2}}\]
\[ \Rightarrow y_2 = \frac{2}{1 - x^2} + \frac{2x \sin^{- 1} x}{\left( 1 - x^2 \right)\sqrt{1 - x^2}}\]
\[ \Rightarrow y_2 = \frac{2}{1 - x^2} + \frac{x y_1}{\left( 1 - x^2 \right)}\]
\[ \Rightarrow y_2 \left( 1 - x^2 \right) = 2 + x y_1 \]
\[ \Rightarrow y_2 \left( 1 - x^2 \right) - x y_1 - 2 = 0\]
Hence proved.
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 \[\tan \left( e^{\sin x }\right)\] ?
Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?
Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
If \[y = \sqrt{a^2 - x^2}\] prove that \[y\frac{dy}{dx} + x = 0\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Differentiate \[\sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x \in R\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
Differentiate
\[\tan^{- 1} \left( \frac{\cos x + \sin x}{\cos x - \sin x} \right), \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?
If \[xy \log \left( x + y \right) = 1\] ,Prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
Differentiate \[{10}^{ \log \sin x }\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \sin x \right)^{\cos x} + \left( \cos x \right)^{\sin x}\] ?
If \[e^x + e^y = e^{x + y}\] , prove that
\[\frac{dy}{dx} + e^{y - x} = 0\] ?
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 \[\frac{dy}{dx}\] when \[x = a \cos \theta \text{ and } y = b \sin \theta\] ?
Find \[\frac{dy}{dx}\], when \[x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)\] ?
Find \[\frac{dy}{dx}\] , when \[x = \cos^{- 1} \frac{1}{\sqrt{1 + t^2}} \text{ and y } = \sin^{- 1} \frac{t}{\sqrt{1 + t^2}}, t \in R\] ?
\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( \frac{1}{2 \sqrt{2}}, \frac{1}{2} \right)\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?
Differentiate \[\sin^{- 1} \left( 2 ax \sqrt{1 - a^2 x^2} \right)\] with respect to \[\sqrt{1 - a^2 x^2}, \text{ if }-\frac{1}{\sqrt{2}} < ax < \frac{1}{\sqrt{2}}\] ?
Differentiate \[\tan^{- 1} \left( \frac{1 - x}{1 + x} \right)\] with respect to \[\sqrt{1 - x^2},\text {if} - 1 < x < 1\] ?
If \[- \frac{\pi}{2} < x < 0 \text{ and y } = \tan^{- 1} \sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}, \text{ find } \frac{dy}{dx}\] ?
If \[y = \sec^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right)\] then write the value of \[\frac{dy}{dx} \] ?
If f (x) is an odd function, then write whether `f' (x)` is even or odd ?
If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .
If y = x + tan x, show that \[\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0\] ?
If x = a (θ + sin θ), y = a (1 + cos θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{a}{y^2}\] ?
Find \[\frac{d^2 y}{d x^2}\] where \[y = \log \left( \frac{x^2}{e^2} \right)\] ?
If y = 500 e7x + 600 e−7x, show that \[\frac{d^2 y}{d x^2} = 49y\] ?
If \[y = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!}\] .....to ∞, then write \[\frac{d^2 y}{d x^2}\] in terms of y ?
Let f(x) be a polynomial. Then, the second order derivative of f(ex) is
If y2 = ax2 + bx + c, then \[y^3 \frac{d^2 y}{d x^2}\] is
If x = sin t and 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 x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
The number of road accidents in the city due to rash driving, over a period of 3 years, is given in the following table:
| Year | Jan-March | April-June | July-Sept. | Oct.-Dec. |
| 2010 | 70 | 60 | 45 | 72 |
| 2011 | 79 | 56 | 46 | 84 |
| 2012 | 90 | 64 | 45 | 82 |
Calculate four quarterly moving averages and illustrate them and original figures on one graph using the same axes for both.
