Good examples of Ansätze

Solution 1:

I am not a great fan of the notion of ansatz, simply because it seems magical.

One can but imagine that most ansätze are the result of whole series of unsuccessful tries, and that we only hear of the successful ones. So whenever I read «let us try the ansatz such and such» I understand «ok, I spent a couple of weeks trying stuff that did not go anywhere but, who knows how, finally managed to make it work, so let me just tell you the short story and, by the by, make myself look like I came up with this stuff out of the blue»

Solution 2:

Separation of variables for linear partial differential equations. The ansatz is the guess that a solution can be written as a tensor product of functions that depend separately on individual coordinate values.

Solution 3:

Perhaps this example is too simple. Think about the wave equation, let's say on the real line: $$ \frac {\partial^2 u}{\partial t^2} = c^2 \frac {\partial^2 u}{\partial x^2}.$$ Let's imagine that we have derived this as the equation governing wave propagation with propagation speed $c$. Physically one expects the simplest solutions to be travelling waves. If $u$ is a wave travelling right (resp left) with speed $c$, then $u$ is a function only of $x-ct$ (resp $x+ct$). So it is meaningful to make the Ansätze $u = f(x-ct)$ and $u = g(x+ct)$ for some $C^2$ functions $f$ and $g$, and once one has the Ansatz it's easy to see that we have now completely solved the equation.