I am trying to solve the following differential equation:
\[ \frac{1}{\sqrt{1+y}} \frac{dx}{dy} - \frac{2\sqrt{1+y}}{x} = 2(x+5). \]

After performing the substitution:
\[ p = \sqrt{1+y}, \quad y = p^2 - 1, \quad dy = 2p , dp, \]
the equation transforms into:
\[ \frac{dx}{dp} = 4p^2\left(\frac{p}{x} + x + 5\right). \]

I attempted the following approaches, but they didn’t work:

1. $Separation of variables:$ The equation mixes $x$ and $p$ in a way that prevents easy separation.
2. $Substitutions:$ I tried substituting $u = x^2$ to simplify the nonlinear terms, but the resulting equation was still complex: \[ \frac{du}{dp} = \frac{4p^3}{u} + 4p^2 + 20p^2\sqrt{u}. \] However, I couldn't find a way to proceed further.

I suspect that the solution might involve more advanced techniques, possibly using special functions or another substitution that I haven’t considered yet.

Could someone guide me on how to approach this equation or suggest resources for solving nonlinear differential equations of this type?

Any help or hints would be greatly appreciated!

I am trying to solve the following differential equation:
\[ \frac{1}{\sqrt{1+y}} \frac{dx}{dy} - \frac{2\sqrt{1+y}}{x} = 2(x+5). \]

After performing the substitution:
\[ p = \sqrt{1+y}, \quad y = p^2 - 1, \quad dy = 2p , dp, \]
the equation transforms into:
\[ \frac{dx}{dp} = 4p^2\left(\frac{p}{x} + x + 5\right). \]

I attempted the following approaches, but they didn’t work:

1. Separation of variables: The equation mixes $x$ and $p$ in a way that prevents easy separation.
2. Substitutions: I tried substituting $u = x^2$ to simplify the nonlinear terms, but the resulting equation was still complex: \[ \frac{du}{dp} = \frac{4p^3}{u} + 4p^2 + 20p^2\sqrt{u}. \] However, I couldn't find a way to proceed further.

I suspect that the solution might involve more advanced techniques, possibly using special functions or another substitution that I haven’t considered yet.

Could someone guide me on how to approach this equation or suggest resources for solving nonlinear differential equations of this type?

Any help or hints would be greatly appreciated!