How Do You Solve This Challenging ODE Involving a Constant?

[bomba923]

I'm a HS senior, and have never covered differential equations outside a CalculusII curriculum. This is not homework, rather only a part (just one part!) of an interesting problem I posed myself (for myself) one day:

How would I solve for $$y(x)$$ from :shy::

$$\frac{{dy}}{{dy}} = \frac{x}{{f - y + \sqrt {x^2 + \left( {f - y} \right)^2 } }}$$

where "f" is a constant

 

[saltydog]

I'm a HS senior, and have never covered differential equations outside a CalculusII curriculum. This is not homework, rather only a part (just one part!) of an interesting problem I posed myself (for myself) one day:

How would I solve for $$y(x)$$ from :shy::

$$\frac{{dy}}{{dy}} = \frac{x}{{f - y + \sqrt {x^2 + \left( {f - y} \right)^2 } }}$$

where "f" is a constant

Hello Bomba. May I suggest you use standard nomenclature: Reserve the 'f' for a function and just use 'k' for a constant:

$$\frac{dy}{dx}=\frac{x}{(k-y)+\sqrt{x^2+(k-y)^2}}$$

Note that the (k-y) expression occurrs twice. How about using a change of variables for starters:

$$u=k-y$$

Do that, then note that even though you have a square root, the equation is still homogeneous of degree 1. Such equations are normally solved by letting:

$$u=xv$$

Make that substitution, do the algebra, simplify, separate variables, integrate, switch back to u, then back to y.