Question

If \( \frac{d}{dx}(y) = y \sin 2x \) and \( y(0) = 1 \), then what is the required solution?

• A. \( y = e^{\cos x} \)
• B. \( y = e^{(\cos 2x)} \)
• C. \( y = e^{\sin x} \)
• D. \( y = 4 \sin x e^{\cos x} \)

Hint:

For separable differential equations, separate the variables and integrate to solve for the unknown function.

Answer 1

The Correct Option is B

Step 1: Apply the differential equation.
Given \( \frac{d}{dx}(y) = y \sin 2x \), we recognize a separable differential equation. Divide by \( y \) and integrate:\[\frac{1}{y} \frac{d}{dx}(y) = \sin 2x\]Integrating yields:\[\ln y = -\frac{1}{2} \cos 2x + C\]Step 2: Determine \( y \).
Exponentiate both sides:\[y = e^{-\frac{1}{2} \cos 2x + C} = e^{C} e^{-\frac{1}{2} \cos 2x}\]Using \( y(0) = 1 \), we get \( e^{C} = 1 \), thus \( y = e^{\cos 2x} \). Final Answer: \[ \boxed{y = e^{\cos 2x}} \]