First slide

Methods of solving first order first degree differential equations

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

Moderate

A

0
B

1
C

2/3
D

3/2

Solution

$\begin{array}{l} y = u^{m} \Rightarrow \frac{d y}{d x} = m u^{m - 1} \cdot \frac{d u}{d x} \\ ∴ Given equation \Rightarrow 2 x^{4} u^{m} \cdot m u^{m - 1} \cdot \frac{d u}{d x} + u^{4 m} = 4 x^{6} \\ \Rightarrow \frac{d u}{d x} = \frac{4 x^{6} - u^{4 m}}{2 m x^{4} u^{2 m - 1}} \end{array}$
For homogeneous equation degree should be same in numerator and denominator
$∴ 6 = 4 m = 4 + 2 m - 1 \Rightarrow m = 3 / 2$

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