Divide and Conquer
If you can move all the \(y\)'s to one side and all the \(x\)'s to the other, the equation is Separable. You solve it by integrating both sides.
\[g(y) dy = f(x) dx \implies \int g(y) dy = \int f(x) dx\]
Worked Examples
Example 1: Basic Separation
Solve \(\frac{dy}{dx} = xy\).
- Separate: \(\frac{1}{y} dy = x dx\).
- Integrate: \(\ln|y| = \frac{x^2}{2} + C\).
- Solve for \(y\): \(y = e^{x^2/2 + C} = Ae^{x^2/2}\).
- Result: \(y = Ae^{x^2/2}\).
The Bridge to Quantum Mechanics
Separation of variables is the most common technique for solving the Schrödinger Equation. We often "separate" the time part from the space part (\(\psi(x, t) = \phi(x)T(t)\)) or the radial part from the angular part in an atom. This breaks one giant, impossible equation into several smaller, solvable ones. This mathematical "divide and conquer" is why we can talk about "Energy States" (\(n\)) and "Orbitals" (\(l, m\)) as separate physical concepts.