Question

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

Answer 1

Answer: 1.5

$y=u^m\Rightarrow \frac{dy}{dx}=m{u}^{m-1}\frac{dy}{dx}.$
Hence $2x^4\cdot u^m\cdot m{u}^{m-1}\cdot \frac{du}{dx}+u^{4m}=4x^6.$
$\frac{du}{dx}=\frac{4x^6-u^{4m}}{2m{x}^4{u}^{2m-1}}$
$\Rightarrow 4m=6$
$\Rightarrow m=\frac{3}{2}$
and $2m-1=2$
$\Rightarrow m=\frac{3}{2}$