Can we solve for $x(t)$ and $y(t)$ in closed-form with respect to time?
Answers
Answered by
0
I'm particularly fascinated by transcendental equations whose posses closed form solutions and when I pose some of them to my friends or teachers I heard a lot of "You can't solve this in closed form" or "You can't solve this with ordinary functions" and I always thought: when and how can I be sure that this is the case ?
In other words suppose that I have an equation in one variable involving some exponentials and polynomials or even trigonometric functions etc. 99% of the time if the solution is nontrivial no one can get to an explicit formula but only to an approximation. But shouldn't we actually have to prove that ordinary functions is not enough to solve such an equation ? I mean may there is a solution which is 3 pages long but contains only common functions.
So what I want to know is:
1) If I have an equation how do I know that I can't solve it using any finite composition of polynomials, exponentials, trigonometrics and their inverses ?
2) If I'm in the case in which I can't and I decide to add one new function to the list (take Lambert's as an example) can I apply a similar reasoning to know when I can use this new function?
clarification for 2) take as an example the equation xex=1xex=1 and let the Lambert's function represent its solution: x=W(1)x=W(1) if then I want to solve x2ex=1x2ex=1 I don't need to introduce another function to my list, the solution still can be expressed in terms of Lambert's function: x=2W(12)x=2W(12) but if I then approach xex+x=1xex+x=1I need another function? How can I prove/disprove this?
In other words suppose that I have an equation in one variable involving some exponentials and polynomials or even trigonometric functions etc. 99% of the time if the solution is nontrivial no one can get to an explicit formula but only to an approximation. But shouldn't we actually have to prove that ordinary functions is not enough to solve such an equation ? I mean may there is a solution which is 3 pages long but contains only common functions.
So what I want to know is:
1) If I have an equation how do I know that I can't solve it using any finite composition of polynomials, exponentials, trigonometrics and their inverses ?
2) If I'm in the case in which I can't and I decide to add one new function to the list (take Lambert's as an example) can I apply a similar reasoning to know when I can use this new function?
clarification for 2) take as an example the equation xex=1xex=1 and let the Lambert's function represent its solution: x=W(1)x=W(1) if then I want to solve x2ex=1x2ex=1 I don't need to introduce another function to my list, the solution still can be expressed in terms of Lambert's function: x=2W(12)x=2W(12) but if I then approach xex+x=1xex+x=1I need another function? How can I prove/disprove this?
Answered by
1
yes we can solve for $x(t)$ and $y(t)$ in closed-form with respect to time.
Similar questions