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

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

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.

__________________________________________________ ___

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 }

__________________________________________________ ____

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.

2. Hint :

$\displaystyle\int \dfrac{dx}{\sqrt{a+x^2}}=\ln |x+\sqrt{a+x^2}|+C,\quad (a\neq 0)$

Fernando Revilla

3. integral { dx/sqrt(a^2 + x^2) } can be found by substituting x as atanu.. so dx = asec^2(u)du,
so you get integral integral (cot^2(u) + 1) du and cot^2 u = cosec^2 -1 ... integral cosec^2(u) du = -cotu ... now substitute back