Advertisements
Advertisements
प्रश्न
If y = 3 e2x + 2 e3x, prove that \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?
Advertisements
उत्तर
Here,
\[y = 3 e^{2x} + 2 e^{3x} \]
\[\text {Differentiating w . r . t . x, we get }\]
\[\frac{d y}{d x} = 6 e^{2x} + 6 e^{3x} \]
\[\text { Differentiating again w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = 12 e^{2x} + 18 e^{3x} \]
\[ \Rightarrow \frac{d^2 y}{d x^2} = 5\left( 6 e^{2x} + 6 e^{3x} \right) - 6\left( 3 e^{2x} + 2 e^{3x} \right)\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = 5\left( \frac{dy}{dx} \right) - 6y\]
\[ \Rightarrow \frac{d^2 y}{d x^2} - 5\left( \frac{dy}{dx} \right) + 6y = 0\]
APPEARS IN
संबंधित प्रश्न
Differentiate sin (3x + 5) ?
Differentiate \[\sqrt{\frac{a^2 - x^2}{a^2 + x^2}}\] ?
Differentiate \[\log \left( \frac{\sin x}{1 + \cos x} \right)\] ?
Differentiate \[\frac{e^x \log x}{x^2}\] ?
Differentiate \[\frac{x^2 + 2}{\sqrt{\cos x}}\] ?
Differentiate \[\log \left( \cos x^2 \right)\] ?
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\], prove that \[2 x\frac{dy}{dx} = \sqrt{x} - \frac{1}{\sqrt{x}}\] ?
If \[y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}\] , prove that \[\left( 1 - x^2 \right) \frac{dy}{dx} = x + \frac{y}{x}\] ?
Differentiate \[\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sin x}{1 + \cos x} \right), - \pi < x < \pi\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
If the derivative of tan−1 (a + bx) takes the value 1 at x = 0, prove that 1 + a2 = b ?
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 \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?
Differentiate \[e^{x \log x}\] ?
Differentiate \[x^{\sin^{- 1} x}\] ?
Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
If \[y^x = e^{y - x}\] ,prove that \[\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\] ?
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 \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\] \[x \in \left( - 1, 0 \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{1 + ax}{1 - ax} \right)\] with respect to \[\sqrt{1 + a^2 x^2}\] ?
If \[f\left( x \right) = x + 1\] , then write the value of \[\frac{d}{dx} \left( fof \right) \left( x \right)\] ?
If f (x) = logx2 (log x), the `f' (x)` at x = e is ____________ .
The derivative of the function \[\cot^{- 1} \left| \left( \cos 2 x \right)^{1/2} \right| \text{ at } x = \pi/6 \text{ is }\] ______ .
If \[y = \left( 1 + \frac{1}{x} \right)^x , \text{then} \frac{dy}{dx} =\] ____________.
If \[y = \sqrt{\sin x + y},\text { then } \frac{dy}{dx} =\] __________ .
If \[f\left( x \right) = \left| x - 3 \right| \text { and }g\left( x \right) = fof \left( x \right)\] is equal to __________ .
If y = e−x cos x, show that \[\frac{d^2 y}{d x^2} = 2 e^{- x} \sin x\] ?
If x = a sec θ, y = b tan θ, prove that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?
If y = cos−1 x, find \[\frac{d^2 y}{d x^2}\] in terms of y alone ?
If \[y = e^{a \cos^{- 1}} x\] ,prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - a^2 y = 0\] ?
If x = 2 cos t − cos 2t, y = 2 sin t − sin 2t, find \[\frac{d^2 y}{d x^2}\text{ at } t = \frac{\pi}{2}\] ?
If \[y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x, \text { then find } \left( x^2 - 1 \right) y_2 + x y_1 =\] ?
If y2 = ax2 + bx + c, then \[y^3 \frac{d^2 y}{d x^2}\] is
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
If p, q, r, s are real number and pr = 2(q + s) then for the equation x2 + px + q = 0 and x2 + rx + s = 0 which of the following statement is true?
