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

Question

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 (A) m=0 (B) m=1 (C) m=(3/2) (D) m=(2/3)

Tardigrade Admin · Accepted Answer

y=um ⇒ (d y/d x)=m ⋅ um-1 ⋅ (d u/d x) Hence, 2 x4 ⋅ m um-1 um ⋅ (d u/d x)+u4 m=4 x6 ⇒ (d u/d x)=(4 x6-u4 m/2 m x4 ⋅ u2 m-1) ∴ 4 m=6 ⇒ m=(3/2)

Q. 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

$m = 0$

$m = 1$

$m = \frac{3}{2}$

$m = \frac{2}{3}$

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

Q. The real value of m for which the substitution y=um will transform the differential equation 2x4⋅ydxdy​+y4=4x6 into a homogeneous equation, is

m=0

m=1

m=23​

m=32​