So as Im sure some of the more mathematically inclined people here know all second order linear differential equations have a mechanical counter part and thus an electrical one as well for vibrations. I am much more familiar with the mechanical side of things.
y''[x]-2*y'[x]+y[x]==Exp[x]/(x^2),y[x],x excuse the mathematica type but it makes it easier to read.
if you were to solve this equation for its solutions you would get
Exp[x]C1+Exp[x]xC2-Exp[x]*(1+Log[x])
That equation baffles my mind. I had no idea how to solve it by hand until I realized that you needed a log equation in there. Never before have I come across any equation with Log in it. The reason why this is interesting is bc it hints at the idea of the Memristor being a real thing which is the fourth and final part to passive components. Ive always been under the idea that the inductor(why is this not a word, plz fix) made up the fourth part with its imaginary outputs but maybe not. As I said Im much more in tune with the mechanical side of things.
Ive never seen a second order ODE spit out a Log and quite frankly I have no idea what that means for electrical systems. And so my question is has anyone else? Or does anyone even have an idea as to how this type of circuit would work or look like? Bc never in any of my years of doing countless differential equations have I seen a log as a solution to this.