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Obtain the Differential Equation by Eliminating Arbitrary Constants A, B from the Equation - Y = a Cos (Log X) + B Sin (Log X) - Mathematics and Statistics

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Obtain the differential equation by eliminating arbitrary constants A, B from the equation -
y = A cos (log x) + B sin (log x)

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Solution

`y = Acos(logx)+Bsin(logx)`

Diff. w.r.t x

`dy/dx=-A(sin(logx))/x+B(cos(logx))/x`

`dy/dx=(-Asin(logx)+Bcos(logx))/x`

`x.dy/dx=-Asin(logx)+Bcos(logx)`

Again diff. w.r.t. x

`x.(d^2y)/(dx^2)+dy/dx=-A(cos(logx))/x-B(sin(logx))/x`

`x.(d^2y)/(dx^2)+dy/dx=-(Acos(logx)+Bsin(logx))/x`

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

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

Concept: Formation of Differential Equation by Eliminating Arbitary Constant
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