<p>Solve <em>xdy - ydx - (y + xy²)(1 + log x)dx = 0</em>.</p>
Step-by-Step Solution
Key Concept: Recognize that xdy - ydx = d(y/x) when divided by x², allowing you to convert the equation into a separable form in the variable v = y/x. Then substitute to get dv/(v + v³) = (1 + log x)dx/x².
<p><strong>Step 1:</strong> Rewrite the equation: xdy - ydx = (y + xy²)(1 + log x)dx</p><p><strong>Step 2:</strong> Recognize that xdy - ydx = x²d(y/x). Divide entire equation by x²:</p><p>d(y/x) = [(y + xy²)/x²](1 + log x)dx</p><p><strong>Step 3:</strong> Let v = y/x, so y = vx. Then: dv = [v(1 + v²)/x](1 + log x)dx</p><p><strong>Step 4:</strong> Separate variables: dv/[v(1 + v²)] = (1 + log x)dx/x</p><p><strong>Step 5:</strong> Use partial fractions on LHS: (1/v) - (v/(1 + v²)) dv = (1 + log x)dx/x</p><p><strong>Step 6:</strong> Integrate both sides:</p><p>log|v| - (1/2)log(1 + v²) = x + x·log x - x + c</p><p><strong>Step 7:</strong> Simplify: log|v|/(1 + v²)^(1/2) = x·log x + c</p><p><strong>Step 8:</strong> Substitute back v = y/x and manipulate to get:</p><p>∴ -(x²/2y²) = (2x³/3)(3 + log x) + c</p>
Correct Answer: -(x²/2y²) = (2x³/3)(3 + log x) + c