If $y=\sin(\log x)$, prove that $x^{2}\dfrac{d^{2}y}{dx^{2}}+x\dfrac{dy}{dx}+y=0$.

CBSE Class 12 · 2007 · DifferentiationLong Answer

y = sin (lo g x)

, prove that

x^{2} \frac{d ^{2} y}{d x ^{2}} + x \frac{d y}{d x} + y = 0

Answer: Proved

Differentiate once:

\frac{d y}{d x} = cos (lo g x) \cdot \frac{1}{x} \Rightarrow x \frac{d y}{d x} = cos (lo g x)

Differentiate this relation again (product rule on the left):

x \frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = - sin (lo g x) \cdot \frac{1}{x}

Multiply by

x

and use

y = sin (lo g x)

x^{2} \frac{d ^{2} y}{d x ^{2}} + x \frac{d y}{d x} = - sin (lo g x) = - y

x^{2} \frac{d ^{2} y}{d x ^{2}} + x \frac{d y}{d x} + y = 0

CBSE Class 12 Board · 2007 · Subjective