Differential Equations
Bernoulli Equations
Grade 12
Question:
<p>The solution of <span class="formula">{xy (1 + \cos x) - y} dx + x dy = 0</span> is</p>
<p>(a) <span class="formula">\frac{x^3}{3} - x^2 \sin x + x \cos x - 2 \sin x + C</span></p>
<p>(b) <span class="formula">\frac{x^3}{3} + x^2 \sin x + 2x \cos x - 2 \sin x + C</span></p>
<p>(c) <span class="formula">\frac{x^3}{3} + 2x^2 \sin x - x \cos x + \sin x + C</span></p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Rewrite the given differential equation in standard form and use substitution to transform it into a linear differential equation. The transformation u = 1/y² simplifies the equation.
<p><strong>Solution:</strong> The given equation can be written as:</p><p><span class="formula">\frac{dy}{dx} + y(1 + \cos x) - \frac{y}{x} = 0</span></p><p>or <span class="formula">\frac{1}{y}\frac{dy}{dx} - \frac{1}{x} = -(1 + \cos x)</span></p><p>Using the transformation <span class="formula">u = \frac{1}{y^2}</span>, the differential equation is solved to get:</p><p><span class="formula">\frac{x^3}{3} + x^2 \sin x + 2x \cos x - 2 \sin x + C</span></p><p>∴ Answer is (b).</p>
Correct Answer: B