Mathematics

This problem asks you to transform a differential equation using a substitution, which changes how derivatives behave. When you substitute $x = \sin t$, the chain rule creates relationships between $\frac{d}{dx}$ and $\frac{d}{dt}$ that simplify the original expression. The key is carefully applying the chain rule twice to rewrite $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ in terms of derivatives with respect to $t$, then substitute everything back into the given expression to reveal what $\lambda$ must be.

This problem asks you to transform a differential equation using a substitution, which changes how derivatives behave. When you substitute $x = \sin t$, the chain rule creates relationships between $\frac{d}{dx}$ and $\frac{d}{dt}$ that simplify the original expression. The key is carefully applying the chain rule twice to rewrite $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ in terms of derivatives with respect to $t$, then substitute everything back into the given expression to reveal what $\lambda$ must be.