I Converting between λ and ν for Blackbody Radiation?!

pl. give the full expression of the quoted equation and where these two are being used.

 

The Rayleigh–Jeans law agrees with experimental results at large wavelengths (low frequencies) but strongly disagrees at short wavelengths (high frequencies). This inconsistency between observations and the predictions of classical physics is commonly known as the ultraviolet catastrophe,

see the full expressions
associated Rayleigh–Jeans limits are given by


or


now you can see the approximations- actually they are not exact expressions

 

Did you look at that Wikipedia page closely? There is a section about the consistency between those two forms.

 

You only have to remember that these are probability distributions ##\mathrm{d} N/\mathrm{d} \nu## or ##\mathrm{d} N/\mathrm{d} \lambda##. Now you have ##\nu=c/\lambda##. This implies
$$B_{\nu}=\frac{\mathrm{d} N}{\mathrm{d} \nu}=\frac{\mathrm{d} N}{\mathrm{d} \lambda} \left|\frac{\mathrm{d} \lambda}{\mathrm{d} \nu}\right| = B_{\lambda} \frac{c}{\nu^2}.$$
Now with
$$B_{\lambda}=\frac{2 c k_B T}{\lambda^4}=\frac{2 k_B T \nu^4}{c^3} \; \Rightarrow\; B_{\nu}=\frac{2 k_B T \nu^2}{c^2},$$
and this was to be shown.