? solution of ( dy / dx ) + ( y / (sqrt(a+(x^2))) ) = 0

smslca

If y = f(x) , and ( dy / dx ) + ( y / ( sqrt(a+(x^2)) ) ) = 0

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

can any one 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 existance 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.

seandoiler

i think you should look up what an integrating factor for a first order linear ODE is

suppose we have [latex] $y'(x)+p(x)y(x)=0$ [/latex], then the integrating factor would be [latex]$u(x)=e^{-\int p(x) dx}[/latex], multiply across the equation by this yields (writing u(x) as integrating factor) [latex]\frac{d(u(x)y)}{dx}=0[/latex] and then just solve this simple derivative equation...

in your case you just need to find out what [latex] \int \frac{1}{\sqrt{a + x^2}} dx [/latex] (that should say a+x^2 under the square root sign...not coming out properly for some reason) is....I'd say have a look in your log tables and check to see if you can find something familiar...

let me know if you're still stuck after that