In how many different ways can this problem be solved?

Question

I have a math problem. In many different ways can this problem be solved? Here is the problem:

$$y''-y'-2y=0, \\ y(0)=1 \\ y'(0)=0$$

I have already found $5$ ways:

$(1):$ Characteristic equation (standard)

$(2):$ Laplace Transforms

$(3):$ Series Solution

$(4):$ Numerical approximation

$(5):$ Reduce to autonomous linear system

Need I say more?

Answer 1

Differential Operators:

$$(D^2-D-2)y=0$$ $$(D-2)(D+1)y=0$$ and the solution follows.

Random Substitutions:

Let $u=y'-2y$. Note that $$y''-y'-2y=y''-2y'+y'-2y=u'+u=0$$ $$u'=-u$$ $$u=y'-2y=c_1e^{-x}$$ and now use integrating factor to solve the first-order problem.

I'll add more as I think of them.