Mathematics

This problem is about recognizing a standard derivative form. The key insight is that $\frac{1}{\sqrt{1-x^2}}$ is the derivative of the inverse sine function—specifically, $\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}}$. Once you identify this, you can integrate directly and use the initial condition $y(0)=3$ to find the constant of integration, then evaluate at $x=\frac{1}{2}$.

This problem is about recognizing a standard derivative form. The key insight is that $\frac{1}{\sqrt{1-x^2}}$ is the derivative of the inverse sine function—specifically, $\frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}}$. Once you identify this, you can integrate directly and use the initial condition $y(0)=3$ to find the constant of integration, then evaluate at $x=\frac{1}{2}$.