I can't seem to understand the use of termination to make the series solutions physically acceptable (when solving the linear harmonic oscillator etc.). So what if the series does not terminate, it's still solving the differential equation, right? Moreover, why is our differential equation yielding non-physical answers?
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1$\begingroup$ I think I understand what you're asking, but an example would help. $\endgroup$– DanielSankCommented Apr 22, 2015 at 3:25
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$\begingroup$ It might depend on the asymptotics. For example, $\sin(x)$ is a function that is bounded by $\pm 1$. It also has an infinite power series expansion in odd powers of $x$, $\sin(x) \sim x - x^3/3 + ...$. Imagine if you cut off the series past some number of terms, say $N$. Then for large $x$ you'd have $\sin(x) \sim x^{2N-1}$ which diverges. In this example the termination is artificial, but if it occurred as a solution, it might rule out a solution as being unphysical as a result of the asymptotic behaviour. $\endgroup$– Surgical CommanderCommented Apr 22, 2015 at 4:31
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So if I understand your question correctly, you are saying that every solution to a mathematically formulated physical problem should necessarily correspond a physical reality. Let me give you a very elementary example. Suppose I give you the following problem 'Person X and person Y are brothers. Y is younger to X by 6 years and the product of their ages is 40.' If you write down the equation of this problem, (taking the age of X to be 'x') you would write $x(x-6)=40$ giving you the quadratic equation $x^{2}-6x-40=0$ . The solution for this equation is $x=10 \space or \space x=-4$ . But now you argue that the age of a person cannot be negative and hence drop '-4' and consider $x=10$ as your answer. So, the point I want to make is that we do this all the time (taking mathematical solution that only corresponds to what we want) . I don't know if, for example in the problem that I mentioned above, '-4' for age 'means' anything philosophically, but I can say for sure that physically it is meaningless. As you continue studying physics you would come across many such instances where we drop terms that don't correspond to what we observe (the most dramatic would be the infinities that we throw out in Quantum Field Theory!) . You should understand that the main point of formulating Physical theories is to understand nature better and if the theory isn't consistent with observations, then maybe the theory will be more relevant when we expand the scope of our experiments (and that way maybe you can argue that nothing is impossible!) , but for now we can neglect those terms which don't match our observations.
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$\begingroup$ Awesome answer! It just clicked, we do this all the time! Thank you! $\endgroup$– MirCommented Apr 22, 2015 at 6:34
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$\begingroup$ I just looked back at this question. I feel like the bounds should come earlier in the process, should we have baked the bounds like 'age can't be negative' into the mathematical model itself? If so does it mean there is a fault in the construction of our model? i.e. Can you think of a way the above problem be stated mathematically in such a way as to acknowledge age cant be negative? $\endgroup$– MirCommented Oct 25, 2017 at 20:06