What ansatz means?

Question

I often see (in the case of ODE) : Let $\dot x=f(x)+\varepsilon g(x)$ and ODE. We make the ansatz that $x(t)=Ae^{\frac{g(x)}{\varepsilon }}.$ What does it mean ? In wikipedia is not well explained. Does it mean that "we suppose that $x(t)=Ae^{\frac{g(x)}{\varepsilon }}$" ? If yes, why can we suppose that ?

Answer 1

"Ansatz" is German but is also used in English. It translates to "approach". So basically you make the approach $x(t)=Ae^{\frac{g(x)}{\varepsilon}}$ and then show that this choice of $x$ does indeed solve your ODE.

Answer 2

The Wikipedia article describes an ansatz as "an educated guess that is verified later by its results". I think that this is quite a good description.

Often we know that a solution must have a certain form. Perhaps you have learnt some rules about how to assert particular solutions of nonhomogeneous first order differential equations, e.g. for $y''(x) + 2 y'(x) + 3 y(x) = x^3$ assert $y_p(x) = a x^3 + b x^2 + c x + d$ and determine $a,b,c,d$. This can be considered to be an ansatz.

In other cases we think that a solution probably can be written on a certain form, for example like $y(x) = f(x) e^x$, and that such an assertion will make solving the equation easier. This is another form of ansatz.