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

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

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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