How do I start this question:
$$\frac{d^2y}{dx^2} = yx$$ for a function $y(x)$ which tends to zero as $x \to \pm \infty$.
Show that transform of $\hat{y}(k)$ of $y(x)$ satisfies the first order differential equation: $$\frac{d\hat{y}}{dk} = ik^2 \hat{y}$$
Notice that $$\partial_\omega \int_{-\infty}^{\infty} y(x) e^{-i \omega x} dx = \int_{-\infty}^{\infty} y(x) \partial_\omega e^{-i \omega x} dx$$ because the integral does not depend on $\omega$, and then calculate the inside of the integral.