Why does the general solution of a second order DE contain two arbitrary constants/coefficients? Can someone explain this to me mathematically?
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- Thread Starter
- 03-04-2013 21:38
- Political Ambassador
- 03-04-2013 21:39
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Last edited by Ateo; 03-04-2013 at 21:51.
- 03-04-2013 21:50
(Original post by Ateo)
- 04-04-2013 01:30
Do you mean to the point where I get the auxiliary equation? That is fine, what confuses me is why is it, that when the auxiliary equation has a repeated root, why don't we just solve for one constant? Why is it necessary to have two and why wouldn't the one with repeated roots work in the same way that the one with two distinct roots works?
Starting with we must have .
However, or even will also give me that differential equation.
The constants are important in defining , but they do not affect (apart from ).
The point of the auxiliary equation is to find such constants. The fact that it occasionally finds only one does not mean that one constant will be sufficient.
If, for example, you are given a differential equation involving acceleration, you can solve it to get an equation for displacement. However, without an initial displacement or velocity, the solution is not very useful.
If you are really interested in what it means for the auxiliary equation to have equal roots, consider that the solution is assumed to get the equation. Equal roots tells us that this solution only works for one value of , after which you have to look elsewhere.
Second order differential equations require two constants - that's non-negotiable.