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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

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

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