Separating variables gives us:
Integrating gives us:
We now proceed to integrate the 2 sides separately. That is, we integrate the left side in y only (since after separating the variables we have terms in y and a dy on the left) and we work on the right side in x only (since we have terms in x and a dx only on the right).
For the right hand side involving x, let u = (1 + 4x2), so du = 8x dx and du/8 = x dx.
So the solution is given by:
We could go on to solve this in y, as follows:
Multiply both sides by `−2`:
For convenience, introduce a new variable `K_1 = -4K`, so that we'll have `-2K=K_1/2`. Our solution becomes:
`1/y^2=-1/2sqrt(1+4x^2)+K_1/2 =` ` (K_1 - sqrt(1+4x^2))/2`
Take the reciprocal of both sides:
Then solve for y:
(The constant K1 can be chosen so that the expression in the denominator is real.)