The real value of $m$ for which the substitution $y = u^m$ will transform the differential equation $2x^3y\frac{dy}{dx} + y^4 = 4x^6$ into a homogeneous equation is:
Step-by-Step Solution
Key Concept: Homogeneity requires matching degrees of all terms in numerator and denominator.
Using the substitution $y = u^m$, we get $\frac{dy}{dx} = mu^{m-1}\frac{du}{dx}$. Substituting into $2x^4u^m\frac{du}{dx} + u^{4m} = 4x^6$ yields $\frac{du}{dx} = \frac{4x^6 - u^{4m}}{2mx^4u^{2m-1}}$. For this to be homogeneous, the degrees of $4x^6 - u^{4m}$ and $2mx^4u^{2m-1}$ must be equal, giving $6 = 4m = 4 + 2m - 1$. Solving these constraints yields $m = \frac{3}{2}$.
Correct Answer: 3