<p>The solution of the differential equation \(y\,dx + (x + x^2y)\,dy = 0\) is</p>
<p>\(-\dfrac{1}{xy} = c\)</p>
<p>\(-\dfrac{1}{xy} + \log y = c\)</p>
<p>\(\dfrac{1}{xy} + \log y = c\)</p>
<p>\(\log y = cx\)</p>
Step-by-Step Solution
Key Concept: Recognize this as an exact differential equation by checking ∂M/∂y = ∂N/∂x, where M = y and N = x + x²y. Alternatively, rearrange to separate variables or find an integrating factor that simplifies the equation.
<p><strong>Step 1:</strong> Rewrite the equation: y dx + (x + x²y) dy = 0</p><p><strong>Step 2:</strong> Check exactness: M = y, N = x + x²y. We have ∂M/∂y = 1, ∂N/∂x = 1 + 2xy. Not exact, so find integrating factor.</p><p><strong>Step 3:</strong> Divide the entire equation by x²: (y/x²)dx + (1/x + y)dy = 0</p><p><strong>Step 4:</strong> Now verify exactness with M = y/x², N = 1/x + y: ∂M/∂y = 1/x², ∂N/∂x = -1/x². This suggests rearranging differently.</p><p><strong>Step 5:</strong> Rewrite as: y dx + x dy + x²y dy = 0, which gives d(xy) + x²y dy = 0</p><p><strong>Step 6:</strong> Divide by xy: d(xy)/(xy) + x dy = 0, or d(ln|xy|) + x dy = 0</p><p><strong>Step 7:</strong> Integrating: ln|xy| + (x²y)/2 = C, or equivalently <strong>xy + (x²y²)/2 = C</strong></p><p>∴ Answer: B</p>
Correct Answer: B