Advertisements
Advertisements
प्रश्न
If x = sin t and y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] .
Advertisements
उत्तर
Given:
x = sint and y = sinpt
Differentiating both sides with respect to t, we get
\[\frac{d x}{d t} = \cos t and \frac{d y}{d t} = p\cos pt\]
\[ \Rightarrow \frac{d y}{d x} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{p\cos pt}{\cos t}\]
Differentiating both sides with respect to x, we get
\[\frac{d^2 y}{d x^2} = \frac{- p^2 \sin pt \cos t + p\cos pt\sin t}{\cos^2 t} \times \frac{dt}{dx}\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = \frac{- p^2 \sin pt \cos t + p\cos pt\sin t}{\cos^3 t}\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = \frac{- p^2 \sin pt \cos t}{\cos^3 t} + \frac{p\cos pt\sin t}{\cos^3 t}\]
\[\Rightarrow \frac{d^2 y}{d x^2} = \frac{- p^2 y}{\cos^2 t} + \frac{x\frac{d y}{d x}}{\cos^2 t}\]
\[ \Rightarrow \cos^2 t\frac{d^2 y}{d x^2} = - p^2 y + x\frac{d y}{d x}\]
\[ \Rightarrow \left( 1 - \sin^2 t \right)\frac{d^2 y}{d x^2} = - p^2 y + x\frac{d y}{d x}\]
\[ \Rightarrow \left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{d y}{d x} + p^2 y = 0\]
APPEARS IN
संबंधित प्रश्न
If the function f(x)=2x3−9mx2+12m2x+1, where m>0 attains its maximum and minimum at p and q respectively such that p2=q, then find the value of m.
Differentiate the following functions from first principles eax+b.
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[\log \left( \frac{x^2 + x + 1}{x^2 - x + 1} \right)\] ?
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
If \[y = e^x \cos x\] ,prove that \[\frac{dy}{dx} = \sqrt{2} e^x \cdot \cos \left( x + \frac{\pi}{4} \right)\] ?
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 \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case: \[y^3 - 3x y^2 = x^3 + 3 x^2 y\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^5 + y^5 = 5 xy\] ?
Find \[\frac{dy}{dx}\] in the following case \[e^{x - y} = \log \left( \frac{x}{y} \right)\] ?
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 \[x^{1/x}\] with respect to x.
Find \[\frac{dy}{dx}\] \[y = x^x + \left( \sin x \right)^x\] ?
If \[x^{16} y^9 = \left( x^2 + y \right)^{17}\] ,prove that \[x\frac{dy}{dx} = 2 y\] ?
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 \[e^{x + y} - x = 0\] ,prove that \[\frac{dy}{dx} = \frac{1 - x}{x}\] ?
If \[y = \log\frac{x^2 + x + 1}{x^2 - x + 1} + \frac{2}{\sqrt{3}} \tan^{- 1} \left( \frac{\sqrt{3} x}{1 - x^2} \right), \text{ find } \frac{dy}{dx} .\] ?
Find \[\frac{dy}{dx}\] , when \[x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}\] ?
If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
Let \[\cup = \sin^{- 1} \left( \frac{2 x}{1 + x^2} \right) \text { and }V = \tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text { then } \frac{d \cup}{dV} =\] ____________ .
If \[f\left( x \right) = \left| x^2 - 9x + 20 \right|\] then `f' (x)` is equal to ____________ .
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
If y = ex (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
