Differential Equations
Bernoulli Equations
Grade 12
Question:
<p>The solution of <span class="formula">x\frac{dy}{dx} + y = y^2 \log x</span> is</p>
<p>(a) <span class="formula">C = y(1 + \log x) + xy</span></p>
<p>(b) <span class="formula">1 = y(1 + \log x) + Cxy</span></p>
<p>(c) <span class="formula">1 = y(1 + \log x) - Cxy</span></p>
<p>(d) <span class="formula">C = y(1 + \log x) - xy</span></p>
Step-by-Step Solution
Key Concept: This is a Bernoulli equation. Divide by y² and substitute v = 1/y to convert it into a linear differential equation. Then apply the standard method for solving linear equations.
<p><strong>Solution:</strong> The given equation can be written as:</p><p><span class="formula">\frac{dy}{dx} + \frac{1}{x}y = \frac{y^2 \log x}{x}</span></p><p>Dividing throughout by <span class="formula">y^2</span>:</p><p><span class="formula">\frac{1}{y^2}\frac{dy}{dx} + \frac{1}{xy} = \frac{\log x}{x}</span></p><p>Let <span class="formula">v = \frac{1}{y}</span>, then <span class="formula">\frac{dv}{dx} = -\frac{1}{y^2}\frac{dy}{dx}</span></p><p>The equation becomes a linear equation in v, which when solved yields:</p><p><span class="formula">1 = y(1 + \log x) + Cxy</span></p><p>∴ Answer is (b).</p>
Correct Answer: B