Advertisements
Advertisements
प्रश्न
If y = 3 cos (log x) + 4 sin (log x), prove that x2y2 + xy1 + y = 0 ?
Advertisements
उत्तर
Here,
\[y = 3 \cos\left( \log x \right) + 4 \sin\left( \log x \right)\]
\[\text { Differentiating w . r . t . x, we get }\]
\[ y_1 = - 3\sin\left( \log x \right) \times \frac{1}{x} + 4 \cos\left( \log x \right) \times \frac{1}{x}\]
\[ = \frac{- 3\sin\left( \log x \right) + 4\cos\left( \log x \right)}{x}\]
\[\text { Differentiating again w . r . t . x, we get }\]
\[ y_2 = \frac{\left( \frac{- 3\cos\left( \log x \right)}{x} - \frac{4\sin\left( \log x \right)}{x} \right) \times x - \left\{ - 3\sin\left( \log x \right) + 4\cos\left( \log x \right) \right\}}{x^2}\]
\[ \Rightarrow y_2 = \frac{- 3\cos\left( \log x \right) - 4\sin\left( \log x \right) - \left\{ - 3\sin\left( \log x \right) + 4\cos\left( \log x \right) \right\}}{x^2}\]
\[ \Rightarrow y_2 = \frac{- 3\cos\left( \log x \right) - 4\sin\left( \log x \right) - \left\{ - 3\sin\left( \log x \right) + 4\cos\left( \log x \right) \right\}}{x^2}\]
\[ \Rightarrow y_2 = \frac{- 3\cos\left( \log x \right) - 4\sin\left( \log x \right)}{x^2} - \frac{\left\{ - 3\sin\left( \log x \right) + 4\cos\left( \log x \right) \right\}}{x^2}\]
\[ \Rightarrow y_2 = \frac{- \left\{ 3\cos\left( \log x \right) + 4\sin\left( \log x \right) \right\}}{x^2} - \frac{\left\{ - 3\sin\left( \log x \right) + 4\cos\left( \log x \right) \right\}}{x^2}\]
\[ \Rightarrow y_2 = \frac{- y}{x^2} - \frac{y_1}{x}\]
\[ \Rightarrow x^2 y_2 = - y - x y_1 \]
\[ \Rightarrow x^2 y_2 + y + x y_1 = 0\]
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove that `y=(4sintheta)/(2+costheta)-theta `
Differentiate the following functions from first principles x2ex ?
Differentiate etan x ?
Differentiate \[\sqrt{\frac{1 + x}{1 - x}}\] ?
Differentiate \[\log \left( cosec x - \cot x \right)\] ?
Differentiate \[x \sin 2x + 5^x + k^k + \left( \tan^2 x \right)^3\] ?
If \[y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}\] ,show that \[\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}\] ?
If \[y = \frac{x}{x + 2}\] , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ?
If \[y = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] .prove that \[\frac{dy}{dx} = 1 - y^2\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{1 + \sqrt{1 - x^2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
If \[y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}\] \[\frac{dy}{dx} \] ?
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
If `y=(sinx)^x + sin^-1 sqrtx "then find" dy/dx`
Find \[\frac{dy}{dx}\] \[y = x^{\log x }+ \left( \log x \right)^x\] ?
If \[y^x = e^{y - x}\] ,prove that \[\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\] ?
If \[\left( \cos x \right)^y = \left( \tan y \right)^x\] , prove that \[\frac{dy}{dx} = \frac{\log \tan y + y \tan x}{ \log \cos x - x \sec y \ cosec\ y }\] ?
If \[y^x + x^y + x^x = a^b\] ,find \[\frac{dy}{dx}\] ?
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 \[x = b \sin^2 \theta \text{ and } y = a \cos^2 \theta\] ?
If \[x = e^{\cos 2 t} \text{ and y }= e^{\sin 2 t} ,\] prove that \[\frac{dy}{dx} = - \frac{y \log x}{x \log y}\] ?
If \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?
Differentiate x2 with respect to x3
If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
\[\frac{d}{dx} \left\{ \tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right) \right\} \text { equals }\] ______________ .
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
If y = cos−1 x, find \[\frac{d^2 y}{d x^2}\] in terms of y alone ?
If y = 3 e2x + 2 e3x, prove that \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?
If \[x = 3 \cos t - 2 \cos^3 t, y = 3\sin t - 2 \sin^3 t,\] find \[\frac{d^2 y}{d x^2} \] ?
\[\text { If }y = A e^{- kt} \cos\left( pt + c \right), \text { prove that } \frac{d^2 y}{d t^2} + 2k\frac{d y}{d t} + n^2 y = 0, \text { where } n^2 = p^2 + k^2 \] ?
If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
If y = (sin−1 x)2, then (1 − x2)y2 is equal to
\[\text { If } y = \left( x + \sqrt{1 + x^2} \right)^n , \text { then show that }\]
\[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = n^2 y .\]
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.
