Separation of variables

Remarks

Example for 1: $y' = x y$ gives $\ln\abs{y} = x^2/2 + C$, so $y = A e^{x^2/2}$ with $A \ne 0$, and $y = 0$ comes from the zero of $h(y) = y$. Check: $(A e^{x^2/2})' = x \cdot A e^{x^2/2}$. Example for 2: for the heat equation $u_t = k u_{xx}$ with $k > 0$, the product $X T$ gives $X(x) T'(t) = k X''(x) T(t)$, so $T'/(kT) = X''/X = \lambda$. With $\lambda = -1$ one finds $X = \sin x$ and $T = e^{-kt}$, and $u = e^{-kt} \sin x$ indeed satisfies $u_t = -k e^{-kt}\sin x = k u_{xx}$. Non-example: $y' = x + y$ is not separable, since $x + y = g(x) h(y)$ for all $x, y$ is impossible. Indeed, $x = 0$ gives $y = g(0) h(y)$ for all $y$, so $h(0) = 0$; then $y = 0$ gives $x = g(x) h(0) = 0$ for all $x$, which is false. In the partial differential setting, superposition of product solutions and boundary conditions lead to eigenvalue problems and Fourier series.