Assumption of solutions for differential equations When solving a differential equation (ODE or PDE), we often make an assumption about the solution in advance to facilitate (or enable in the first place) the solution process. Take e. g. an ODE of type $$ay(x)''+by(x)'+cy(x)=0$$ which is (in case the characteristic polynomial has non-identical real roots $k_1$ and $k_2$) solved by a function of type $$y=C_1e^{k_1x}+C_2e^{k_2x}$$ which has been proven to be the "one and only" solution. My question: How was that solution (and the generic solutions of other differential equation types) determined "the first time around"? Is there a way to systematically determine them starting from the original ODE/PDE or did they "reverse engineer" them? (Like hmmm let's try an exponential function. Heureka!)
 A: In the case of linear, constant-coefficient, homogeneous:
$$ay''+by'+cy = 0,$$
one might look at the equation and say, "Hmmm...I need a function that is a multiple of its own derivatives...." and decide $y=e^{rx}$ might be worth a try.  For a homogeneous Euler equation:
$$ax^2y''+bxy'+cy = 0,$$
one might say, "Hmmmm, I need a function that every time I take a derivative, and then multiply by $x$, I get multiple of that function.  So when I take the derivative, I reduce the power of $x$ by one...."  and decide that $y=x^r$ might be a good choice.
In the non-homogeneous linear case, one usually find the homogeneous solution $y_h$ first.  Then he might say, "The homogeneous solution almost solves the equation, and any constant multiple $cy_h$ solves the equation, so maybe if I make the constant a variable, I could get a solution to the non-homogeneous part." And decide that $y=u(x)y_h(x)$ might be worth a try.
So it's not just random guessing, but looking at the equation and deducing properties of the solution.  A little bit if guessing, but a lot of thinking.
