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If y=log[x+sqrt(x^2+a^2)] show that (x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0 - Mathematics

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If `y=log[x+sqrt(x^2+a^2)] ` show that `(x^2+a^2)(d^2y)/(dx^2)+xdy/dx=0`

 
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Solution

 

It is given that:

`y=log[x+sqrt(x^2+a^2)]`

Differentiating equation (1) with respect to x, we get

`dy/dx=(1+x/sqrt(x^2+a^2))/(x+sqrt(x^2+a^2))`

`dy/dx=1/sqrt(x^2+a^2)..........(2)`

`xdy/dx=x/sqrt(x^2+a^2)...........(3)`

Again differentiating equation (2) with respect to x, we get

`(d^2y)/(dx^2)=-x/(x^2+a^2)^(3/2)`

`(x^2+y^2)(d^2y)/(dx^2)=-x/sqrt(x^2+a^2)..............(4)`

Adding equation (3) and (4), we get

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

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

 
Concept: Logarithmic Differentiation
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