▶️ Answer/Explanation
Solution
Separate variables: \( dy = -x \cos(x^2) \, dx \).
Integrate: \(\int dy = \int -x \cos(x^2) \, dx\).
Use substitution \( u = x^2 \), \( du = 2x \, dx \), so \( x \, dx = \frac{du}{2} \).
Then, \(\int -x \cos(x^2) \, dx = -\frac{1}{2} \int \cos(u) \, du = -\frac{1}{2} \sin(u) + C = -\frac{1}{2} \sin(x^2) + C\).
So, \( y = -\frac{1}{2} \sin(x^2) + C \).
Apply initial condition \( y = 2 \) when \( x = 0 \): \( 2 = -\frac{1}{2} \sin(0) + C \), so \( C = 2 \).
Thus, \( y = -\frac{1}{2} \sin(x^2) + 2 \).
Verify: \(\frac{dy}{dx} = -\frac{1}{2} \cos(x^2) \cdot 2x = -x \cos(x^2)\), which matches.
✅ Answer: D