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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

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a

m = 0

b

m = 1

c

m = 3/2

d

m =3

answer is C.

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Detailed Solution

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$y = u^{m} \Rightarrow \frac{d y}{d x} = m u^{m - 1} \frac{d y}{d x}$
Since, $2 x^{4} . u^{m} . m u^{m - 1} . \frac{d u}{d x} + u^{4 m} = 4 x^{6}$
$\frac{d u}{d x} = \frac{4 x^{6} - u^{4 m}}{2 m x^{4} u^{2 m - 1}}$
$\Rightarrow 4 m = 6 \Rightarrow m = \frac{3}{2}$
And $2 m - 1 = 2 \Rightarrow m = \frac{3}{2}$

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The real value of m for which the substitution,y=um will transform the differential equation,2x4ydydx+y4=4x6 into a homogeneous equation is