Question

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

• A. $m=0$
• B. $m=1$
• C. $m=\frac{3}{2}$
• D. $m=\frac{2}{3}$

Answer 1

Answer: (C)

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