If $y=3\cos(\log x)+4\sin(\log x)$, show that $x^{2}\dfrac{d^{2}y}{dx^{2}}+x\dfrac{dy}{dx}+y=0$.

CBSE Class 12 · DifferentiationLong Answer

y = 3 cos (lo g x) + 4 sin (lo g x)

, show 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} = \frac{1}{x} [- 3 sin (lo g x) + 4 cos (lo g x)]

x \frac{d y}{d x} = - 3 sin (lo g x) + 4 cos (lo g x)

. Differentiate again (product rule on the left):

x \frac{d ^{2} y}{d x ^{2}} + \frac{d y}{d x} = \frac{1}{x} [- 3 cos (lo g x) - 4 sin (lo g x)]

Multiply by

x

; note

- 3 cos (lo g x) - 4 sin (lo g x) = - y

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

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

CBSE Class 12 Board · Subjective