<p>The real value of \(m\) for which \(y = x^m\) transforms \(2x^4 y\,\dfrac{dy}{dx} + y^2 = 4x^6\) into a homogeneous equation:</p>
Step-by-Step Solution
Key Concept: Substitute y = x^m, compare degrees to make the equation homogeneous.
<div class='solution'><p><strong>Step 1:</strong> Substitute $y = x^m$, $\dfrac{dy}{dx} = mx^{m-1}$:</p>
<p>$$2x^4 \cdot x^m \cdot mx^{m-1} + x^{2m} = 4x^6$$</p>
<p>$$2mx^{2m+3} + x^{2m} = 4x^6$$</p>
<p><strong>Step 2:</strong> For the equation to become homogeneous (equal degrees or expressible in $y/x$), the $x$-power of the first term must equal the RHS power:</p>
<p>$$2m + 3 = 6 \implies m = \tfrac{3}{2}$$</p>
<p>Check second term: $x^{2m} = x^3$ — this allows the equation to be written in terms of $z = y/x^{3/2}$. <strong>Answer: (B)</strong> $m = 3/2$.</p>
<p class='key-concept'>🔑 Key Concept: For $y = x^m$ to make an equation homogeneous, match the highest degree terms in $x$ after substitution.</p>
<p class='trap-warning'>⚠️ Trap: Setting $2m = 6$ (wrong term matching) and getting $m=3$ instead of correctly solving $2m+3=6$.</p></div>
Correct Answer: 2