If y = 3 cos (log x) + 4 sin (log x), show that x^2(d^2y/dx^2)+x(dy/dx)+y=0. ​

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If y = 3 cos (log x) + 4 sin (log x), show that

$x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+y=0.$

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Given,

y = 3 cos (log x) + 4 sin (log x)

Differentiating with respect to x, we get

$\frac{dy}{dx}=-\frac{3sin(log\,x)}{x}+$$\frac{4cos(log\,x)}{x}$

⇒ y1$\frac{1}{x}$[- 3 sin(log x) +4 cos (log x)]

Again differentiating with respect to x, we get

Now,

LHS = x2y2 + xy1 +y

= ($x^2(\frac{-sin(logx)-7\,cos(logx)}{x^2})$$x\times\frac{1}x$[-3 sin(log x) + 4 cos(log x)] +3 cos (log x) + 4 sin(log x)

=  - sin (log x) - 7 cos (log x) - 3 sin (log x) + 4 sin (log x)

= 0 = RHS

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