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f such that f'' = -ff = A sin(x + B)Find a function g such that g'' = C * -g and
g(0) = Dg'(0) = 0
where C and D are arbitrary constants.For those who have finished the extension
Congratulations, you have just derived the formula for objects in simple harmonic motions (physics).
when i first showed this question to my friend, he told me that f(x) = 0 satisfies the condition. It is incredible that he said this without delay (i have not thought of that possibility at all), but at the same time it made me wanted to explode,
That's why one usually searches for non trivial solution of differential equations. The trivial solution f(x)=0, in physics: "nothing happens", is usually not of interest, as we know by experience that something must happen to a system when some initial action is performed, yet mathematically it is always there.
For simplicity, I restricted my solution to the 'extension' part and embedded the first part into it.
I hope it is OK to upload it here.
The task is a neat example to train the procedure of finding a solution to ODEs with initial value problems.
Thank you!
Solution.
This challenge requires some specific knowledge on the topics, so if you have no idea how to approach this question, take a hint
Initial hint
Maclaurin series: https://openstax.org/books/calculus-volume-2/pages/6-introduction
If you found f but not in the specific format
hint
R formula: https://brilliant.org/wiki/trigonometric-r-method/
No hints on the extension, you are on your own.