<p>The solution of \((2x - 10y^3)\frac{dy}{dx} + y = 0\) is \(xy^2 + ly^5 + C\). Then \(l\) is</p>
Step-by-Step Solution
Key Concept: Rearrange the differential equation to identify exact differentials or integrating factors.
<p>Given: $(2x - 10y^3)\frac{dy}{dx} + y = 0$</p><p>Rearranging: $(2x - 10y^3)dy + y\,dx = 0$</p><p>Or: $2x\,dy + y\,dx - 10y^3 dy = 0$</p><p>Notice that $d(xy) = y\,dx + x\,dy$, but we have $2x\,dy + y\,dx = d(xy) + x\,dy$</p><p>The equation becomes: $d(xy) + x\,dy - 10y^3\,dy = 0$</p><p>Dividing by $y^2$: This is an exact equation.</p><p>Solution: $xy^2 + 5y^5 = C$, so $l = 5$</p>
Correct Answer: C