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Why is the solution in the form of Ce^kx ? 
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#1
Jun2514, 11:57 AM

P: 6

Why is the solution to linear differential equations with constant coefficients sought in the form of Ce^kx ?
I have heard that there is linear algebra involded here. Could you please elaborate on this ? 


#2
Jun2514, 02:21 PM

P: 356

Once you provide initial conditions, those differential equations will define a unique solution (Since in principle you can just numerically integrate it).
So once you have n (the order of the DE) linearly independent solutions (and thus are able to satisfy the initial conditions), you have a unique solution. This is my understanding, there is probably some more technical way to put it. 


#3
Jun2514, 02:49 PM

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P: 6,962

The solution of a firstorder linear DE with constant coefficients is an exponential function follows directly from the definition of an exponential function.
For an n'th order DE, either you can convert it into a system of n firstorder DE's, or the fundamental theorem of algebra says that you can always factorize it as ##(\frac{d}{dx}  a_1)(\frac{d}{dx}  a_2)\cdots(\frac{d}{dx}  a_n)y(x) = 0##. 


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