1. Tardigrade
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  4. The real value of m for which the substitution, y=um will transform the differential equation, 2 x4 y (d y/d x)+y4=4 x6 into a homogeneous equation is :

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Q. 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 :

Differential Equations

Solution:

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