<p>Solve <em>xdx + ydy</em> + <em>(xdy - ydx)/(x² + y²)</em> = 0.</p>
Step-by-Step Solution
Key Concept: Recognize that xdx + ydy = d(x² + y²)/2 and (xdy - ydx)/(x² + y²) = d(arctan(y/x)), then integrate to get a logarithmic-trigonometric relation that yields the implicit solution.
<p><strong>Step 1:</strong> Rewrite the equation by identifying key differentials:</p><p>xdx + ydy = ½d(x² + y²)</p><p>(xdy - ydx)/(x² + y²) = d(arctan(y/x))</p><p><strong>Step 2:</strong> Substitute into the original equation:</p><p>½d(x² + y²) + d(arctan(y/x)) = 0</p><p><strong>Step 3:</strong> Integrate both sides:</p><p>½ln(x² + y²) + arctan(y/x) = k₁ (where k₁ is constant)</p><p><strong>Step 4:</strong> Rearrange to isolate arctan(y/x):</p><p>arctan(y/x) = k₁ - ½ln(x² + y²)</p><p><strong>Step 5:</strong> Set c = 2k₁, then take tangent of both sides:</p><p>y/x = tan((c - x² - y²)/2)</p><p><strong>Step 6:</strong> Solve for y:</p><p>∴ <strong>y = x·tan((c - x² - y²)/2)</strong></p>
Correct Answer: y = x·tan((c - x² - y²)/2)