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If y=2 cos(logx)+3 sin(logx), prove that x^2(d^2y)/(dx2)+x dy/dx+y=0 - Mathematics

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If y=2 cos(logx)+3 sin(logx), prove that `x^2(d^2y)/(dx2)+x dy/dx+y=0`

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Solution

y=2 cos(logx)+3 sin(logx)

Differentiating both sides with respect to x, we get

`dy/dx=2xxd/dx cos(logx)+3xx d/dxsin(log x)`

`=-2sin(logx)xx1/x+3 cos(logx)xx1/x`

`=>x dy/dx=-2 sin(logx)+3 cos(logx)`

Again, differentiating both sides with respect to x, we get

`x (d^2y)/(dx^2)+dy/dx=-2cos(logx)xx1/x-3 sin(logx)xx1/x`

`x^2 (d^2y)/(dx^2)+xdy/dx=-[2 cos(logx)+3sin(logx)]`

`x^2 (d^2y)/(dx^2)+xdy/dx=-y`

`x^2 (d^2y)/(dx^2)+xdy/dx+y=0`

Concept: Second Order Derivative
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