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