Differential Equations
Bernoulli Equations
Grade 12
Question:
<p>The solution of <span>\(\frac{dy}{dx} + 2y\cot x = y^2\)</span> is</p>
<p>(A) <span>\(y = 0\)</span></p>
<p>(B) <span>\(y = \frac{c}{1 + \cos x}\)</span></p>
<p>(C) <span>\(x + 2\sin^{-1}\frac{c}{2y} = 0\)</span></p>
<p>(D) <span>\(x + 2\cos\frac{c}{2y} = 0\)</span></p>
Step-by-Step Solution
Key Concept: Recognize Bernoulli equations and apply the standard substitution to convert to linear form.
<p>This is a Bernoulli equation. Use substitution <span>$v = \frac{1}{y}$</span> to convert it to linear form, then solve the resulting linear differential equation.</p>
Correct Answer: B