The real value of m for which the substitution $y=u^m$ will transform the differential equation  $2x^{4}y\frac{dy}{dx}+y^4=4x^6$ into a homogeneous equation is

$\begin{array}{l}y=u^m\Rightarrow \frac{dy}{dx}=m{u}^{m-1}\cdot \frac{du}{dx}\\ \therefore \text{ Given equation }\Rightarrow 2x^4{u}^m\cdot m{u}^{m-1}\cdot \frac{du}{dx}+u^{4m}=4x^6\\ \Rightarrow \frac{du}{dx}=\frac{4x^6-u^{4m}}{2m{x}^4{u}^{2m-1}}\end{array}$

For homogeneous equation degree should be same in numerator and denominator

$\therefore 6=4m=4+2m-1\Rightarrow m=3/2$