>>12012800>constant coeff linear ODE's where the answer just consists of 'just guess the solution lol'These are the only kind where you don't have to guess the solution... thats why they are always most of the material
>I would understand it if they covered existence theorems in depth but they didn'tWell you don't need to prove a solution exists if you find it, even by guessing. Uniqueness and existance theorems only give you the necessary conditions so that there is a unique solution to the initial value problem. This means there are initial value problems which do not satisfy the conditions of the theorems yet they still have a unique solution. So they help you only in specific cases and only to show that there are no other solutions which you haven't found (or that there is a solution if you haven't found it). On top of that they are very hard to prove
>The laplace part was also rushed and felt like a needless botherIt kind of is though. The only thing you can do more with it is solve equations where there is something like diracs delta or other non continous functions
>>12012932>variation of parametersThe idea is that if the coefficients were constant you would get 0 when plugging it into the equation. If they are not you get some leftovers from c' and you can find the requirements so that the equation is true and get a partial solution
>phase planeYou do realise that this is equivalent to "proof by picture"
>how about giving a real example of a system?Idk why they didn't give you examples (maybe you took a more theoritical class? but then why do you care about applications?). Some examples of systems are two objects connected with a spring, any circuit with parallel components, an objects fall if you consider position and velocity independent varibles of time, a double pendulum, the stock market etc. They aren't really important though because you can apply what you learned wherever you are interested and find your own examples
