Differential Equations
First Order Linear ODEs
Grade 12
Question:
<p>The solution of the differential equation \(\frac{dy}{dx} + \frac{1}{x}y = \frac{1}{x}\cos y \sin y\), where \(y = -1\) as \(x \to \infty\), is</p>
<p>(A) \(y = \sin^{-1}\left(\cos\frac{1}{x}\right)\)</p>
<p>(B) (Incomplete in source)</p>
<p>(C) (Incomplete in source)</p>
<p>(D) (Incomplete in source)</p>
Step-by-Step Solution
Key Concept: Use integrating factor method for linear first-order ODEs and apply boundary condition to determine the constant.
<p>Apply integrating factor method with $IF = e^{\int \frac{1}{x}dx} = x$. Multiply through and recognize the special trigonometric form to solve.</p>
Correct Answer: A