Advertisements
Advertisements
Question
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
Advertisements
Solution
Here,
\[y = 2 \sin x + 3 \cos x\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{d y}{d x} = 2\cos x - 3 \sin x\]
\[\text { Differentiating again w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = - 2 \ sin \ x - 3 \ cos \ x \]
\[ \Rightarrow \frac{d^2 y}{d x^2} = - \left( 2 \sin x + 3 \ cos x \right)\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = - y\]
\[ \Rightarrow \frac{d^2 y}{d x^2} + y = 0\]
Hence proved.
APPEARS IN
RELATED QUESTIONS
Differentiate the following functions from first principles e3x.
Differentiate the following functions from first principles log cosec x ?
Differentiate logx 3 ?
Differentiate \[\log \left( \cos x^2 \right)\] ?
If \[y = e^x \cos x\] ,prove that \[\frac{dy}{dx} = \sqrt{2} e^x \cdot \cos \left( x + \frac{\pi}{4} \right)\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[xy = c^2\] ?
Find \[\frac{dy}{dx}\] in the following case \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?
If \[y = x \sin y\] , Prove that \[\frac{dy}{dx} = \frac{\sin y}{\left( 1 - x \cos y \right)}\] ?
If \[\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find} \frac{dy}{dx}\] ?
If \[\sqrt{y + x} + \sqrt{y - x} = c, \text {show that } \frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2}{x^2} - 1}\] ?
Differentiate \[{10}^\left( {10}^x \right)\] ?
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
If \[x^{16} y^9 = \left( x^2 + y \right)^{17}\] ,prove that \[x\frac{dy}{dx} = 2 y\] ?
If \[y^x = e^{y - x}\] ,prove that \[\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\] ?
If \[e^{x + y} - x = 0\] ,prove that \[\frac{dy}{dx} = \frac{1 - x}{x}\] ?
If \[y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sec^2 x}{2 y - 1}\] ?
If \[y = \left( \tan x \right)^{\left( \tan x \right)^{\left( \tan x \right)^{. . . \infty}}}\], prove that \[\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}\] ?
Differentiate x2 with respect to x3
Differentiate \[\cos^{- 1} \left( 4 x^3 - 3x \right)\] with respect to \[\tan^{- 1} \left( \frac{\sqrt{1 - x^2}}{x} \right), \text{ if }\frac{1}{2} < x < 1\] ?
If f (x) = loge (loge x), then write the value of `f' (e)` ?
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
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 \[\sin y = x \cos \left( a + y \right), \text { then } \frac{dy}{dx}\] is equal to ______________ .
If x = a (θ + sin θ), y = a (1 + cos θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{a}{y^2}\] ?
If x = a(1 − cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{1}{a}\text { at } \theta = \frac{\pi}{2}\] ?
If y = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \cdot \frac{dy}{dx} + y \cos^2 x = 0\] ?
If log y = tan−1 x, show that (1 + x2)y2 + (2x − 1) y1 = 0 ?
If \[y = \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right]^2\] show that \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 2\] ?
\[\text { If y } = x^n \left\{ a \cos\left( \log x \right) + b \sin\left( \log x \right) \right\}, \text { prove that } x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0 \] Disclaimer: There is a misprint in the question. It must be
\[x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0\] instead of 1
\[x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0\] ?
If y = x + ex, find \[\frac{d^2 x}{d y^2}\] ?
If x = f(t) and y = g(t), then write the value of \[\frac{d^2 y}{d x^2}\] ?
If x = a cos nt − b sin nt, then \[\frac{d^2 x}{d t^2}\] is
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If xy − loge y = 1 satisfies the equation \[x\left( y y_2 + y_1^2 \right) - y_2 + \lambda y y_1 = 0\]
Find the minimum value of (ax + by), where xy = c2.
