Differential Equations
Homogeneous Differential Equations
Grade None
Question:
<p>If \(x\dfrac{dy}{dx} = y(\log y - \log x + 1)\), then the solution of the equation is</p>
<p>\(y\log\left(\dfrac{x}{y}\right) = cx\)</p>
<p>\(x\log\left(\dfrac{y}{x}\right) = cy\)</p>
<p>\(\log\left(\dfrac{y}{x}\right) = cx\)</p>
<p>\(\log\left(\dfrac{x}{y}\right) = cy\)</p>
Step-by-Step Solution
Key Concept: Recognize this as a homogeneous differential equation by rewriting it in the form dy/dx = f(y/x). Use the substitution v = y/x to separate variables and integrate.
<p><strong>Step 1:</strong> Rewrite the given equation:</p><p>x(dy/dx) = y(log y - log x + 1)</p><p>dy/dx = (y/x)(log(y/x) + 1)</p><p></p><p><strong>Step 2:</strong> Recognize this is homogeneous. Use substitution v = y/x, so y = vx and dy/dx = v + x(dv/dx)</p><p></p><p><strong>Step 3:</strong> Substitute into the equation:</p><p>v + x(dv/dx) = v(log v + 1)</p><p>x(dv/dx) = v·log v</p><p></p><p><strong>Step 4:</strong> Separate variables:</p><p>(dv)/(v·log v) = dx/x</p><p></p><p><strong>Step 5:</strong> Integrate both sides:</p><p>∫dv/(v·log v) = ∫dx/x</p><p>log|log v| = log|x| + C</p><p></p><p><strong>Step 6:</strong> Simplify:</p><p>log|log v| = log|x| + log|C₁|</p><p>log v = C₁x</p><p></p><p><strong>Step 7:</strong> Substitute back v = y/x:</p><p>log(y/x) = C₁x</p><p>∴ Answer: C</p>
Correct Answer: C