Question

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

Answer 1

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