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I knew its solution is y = { sqrt(a+(x^2)) - x } , where a is a constant

can anyone give the proof , by solving the differntial equation.

Are there any other solutions for the above given differential equation. I asked this other

solutions because, on rearranging the given differential equation we get

( dy / y ) = - ( dx / ( sqrt(a+(x^2)) ) )

on integrating

ln y = { integral ( - ( dx / ( sqrt(a+(x^2)) ) ) ) } + ln c

so I may get the solution as y = ce^y1 , where y1 is funcion of x other than f(x)

I am not sure about the existence of general solution , but I think it may exist.

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so what I want is ,

1. Solving procedure for the differntial equation to get the solution

y = { sqrt(a+(x^2)) - x }

2. what is general answer for { integral ( - ( dx / ( sqrt(a+(x^2)) ) ) }

3. Is there any general solution for the differential equation given.

other than y = { sqrt(a+(x^2)) - x }

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Procedures I have tried and falied to do further.

1. we know d(sqrt(a+(x^2))) / dx = x / sqrt(a+(x^2))

so 1 / sqrt(a+(x^2)) = ( d(sqrt(a+(x^2))) / dx ) /x

on substituting this value in the differential equation , we will get

( dy / dx ) + ( ( y * d( sqrt(a+(x^2)))/dx ) / x ) = 0

on solving this I got strucked at

ln y = ( - ( sqrt(a+(x^2)) ) / x ) - { integral ( sqrt(a+(x^2)) / x^2) dx )

2. rearranging the differntial equation we get

( dy / y ) = - ( dx / ( sqrt(a+(x^2)) ) )

take x = a cos(t)

dx = - a sin(t) dt

t = cos^-1 (x/a)

on solving I got strucked at

( dy / y ) = { ( sqrt(2p) * sin(t) ) / sqrt( cos(2t) + 3 ) } dt

I got no other ideas. I wish , I can get the answers for all the 3 questions I have asked.