I noticed a pattern recently where you can solve problems by "guessing" the form of the solution
then solving a polynomial
Lets say we have a differential equation of the form
Quadratic equation! solve for then use intial conditions to find and
This also shows there is always a solution for a diffeq of the form
Since you can let then factor out and solve the polynomial.
a solution always exists if
Quadratic equation! solve for then use initial conditions to find and .
Sadly we must verify our solution for works, since unlike the differential equations example
we don't know if is true or not.
Exponentiating just raises each then divides and sums them up, so we get
- [x] Justify always being a "pure" solution to a the diffq
- [ ] Justify being a reasonable assumption for solving recurrances
- [ ] Show how second order linear diffq is identical to system of 2 first order diffq
- [ ] Send to euler2 for peer review
Eigenstuff, connection with markov matrices.
is diagonalizing a markov matrix the same as solving the recurrance?