Xy'=y+x^n/cos(y/x). For which value of n this differential equation is with homogeneous? How to solve the DE for any n integer?

Divide both sides by x:
dy/dx = y/x + x^(n-1)/cos(y/x).

This DE is homogeneous iff it can be written in the form
dy/dx = F(y/x) for some function F.

For our DE, this is the case precisely when n-1 = 0, or equivalently n = 1.
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Let y = xv, so that dy/dx = v + x dv/dx.

Then, the DE transforms to
x(v + x dv/dx) = xv + x^n/cos(v)
==> x^2 dv/dx = x^n/cos(v)

Separating variables:
cos v dv = x^(n-2) dx

Integrate both sides:
sin v = x^(n-1)/(n-1) + C if n does not equal 1
or sin v = ln |x| + C if n = 1.

Hence,
sin(y/x) = x^(n-1)/(n-1) + C if n does not equal 1
or sin(y/x) = ln |x| + C if n = 1.
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I hope this helps!