The certainty behind the spanning set of solution for a second order linear homogeneous ODE with constant coefficients I observed that the solution space of ODE $$y''(t)+ay'(t)+by(t)=0, t\in [a,b],$$ with constant real coefficients $a$ and $b$, is always spanned by the functions $y_\lambda(t)=e^{\lambda t}$. But the characteristic equation $$\lambda^2+a\lambda+b=0$$ can only be obtained by substituting $e^{\lambda t}$ to the given ODE. So, how to say all possible solutions (other than polynomials, sinusoidal, exponential (if possible)) for the above ODE can be spanned by these basis vectors? OR how can we reject the possibility of such crazy functions beyond the spanning set as a solution for the same?
 A: There is a very neat way of approaching this that leaves absolutely no doubt of all of the solutions. I don't have a lot of time so I will be schematic about this. In any case, I think everything is found in this lecture https://www.youtube.com/watch?v=tV-xIhP7VU8&list=PLOFVFbzrQ49TNlDOxxCAjC7kbnorAR1MU&index=2
First, as shown in the lecture, you can use an integrating factor to reduce to the case $a=0$. In that case, the remaining differential equation can be factored as
$$\left(\frac{d}{dx}+i\sqrt{b}\right)\left(\frac{d}{dx}-i\sqrt{b}\right)y=0.$$
Setting
$$Y=\left(\frac{d}{dx}-i\sqrt{b}\right)y,$$
you are left with a simple first order equation
$$\left(\frac{d}{dx}+i\sqrt{b}\right)Y=0,$$
which can be rewritten as
$$\frac{d\ln(Y)}{dx}=-i\sqrt{b}.$$
At which point is clear that all solutions are of the form
$$Y(x)=Ae^{-i\sqrt{b}x}$$
for a constant $A$. We are then left with another first order differential equation
$$\left(\frac{d}{dx}-i\sqrt{b}\right)y=Ae^{-i\sqrt{b}x},$$
which can be solved by using an integrating factor. In particular, the solution to that will have another arbitrary constant and we see that the space of solutions is two dimensional.
This procedure is very neat and general. Moreover, it doesn't rely on any unnecessary ansatz. It shows the importance of integrating factors, which, as explained in the lectures, are just the study of the behavior of the differential equation under local scale transformations, and factoring differential operators, which has been an extremely important subject in math (see Dirac equation, or the work of David Eisenbud https://www.youtube.com/watch?v=wTUSz-HSaBg)
