Why is the solution in the form of Ce^kx ?

[joo]

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 ?

 

[HomogenousCow]

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.

 

[AlephZero]

The solution of a first-order 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 first-order DE's, or the fundamental theorem of algebra says that you can always factorize it as $(\frac{d}{dx} - a_1)(\frac{d}{dx} - a_2)\cdots(\frac{d}{dx} - a_n)y(x) = 0$.