To get W|A to recognize things properly, give your y functions explicit arguments as in

y(x)"+(a-(sqrt(c+y(x)^2)))y(x)=0

or

y[x]"+(a-(sqrt(c+y[x]^2)))y[x]=0

it may well be that your differential equation does not have an explicit closed form solution. In Mathematica trying

DSolve[y''[x] + (1 - Sqrt[1 + y[x]^2]) y[x] == 0, y[x], x]

grinds along for some while--it hasn't yet spit out an answer after several minutes.

The first order version of this (i.e., only a single derivative for the first term),

DSolve[y[x] + (1 - Sqrt[1 + y[x]^2]) y[x] == 0, y[x], x]

which is trivially integrable yields

{{y[x] -> 
   InverseFunction[-((
       1 - Log[#1] #1^2 + Log[1 + Sqrt[1 + #1^2]] #1^2 + Sqrt[
        1 + #1^2])/(2 #1^2)) &][x + C[1]]}}

because the integration cannot be analytically inverted to get y in terms of x.

Ok, now the second order problem yielded a solution: here it is:

Solve[Integrate[1/Sqrt[C[1] + 2*((-(1/2))*K[1]^2 + (1/3)*(1 + K[1]^2)^(3/2))], 
     {K[1], 1, y[x]}]^2 == (x + C[2])^2, y[x]]

I.e. it is implicit and in terms of inverting a function expressed as an integral that Mathematica does not know how to do. I.e.,

Integrate[1/Sqrt[c + 2*(-x^2/2 + (1 + x^2)^(3/2)/3)], x]