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Suppose we have the following nth order homogeneous linear differential equation, $$a_0\frac{d^ny}{dx^n}+a_1\frac{d^{n-1}y}{dx^{x-1}}+...+a_{n-1}\frac{dy}{dx}+a_ny=0$$, $a_0 \neq 0$ Suppose that we have the coefficients which are real such that $a_0,...,a_n$. We can have $y=e^{mx}$ provided $e^{mx} \neq0$ as a solution to the differential equation. This is because $\frac{d^k f(x)}{dx^k}=m^ke^{mx}$ where $k=[0,1,2,...,n]$

With the above saying hold we would have auxiliary equation or sometimes it is known as the characteristic equation!

$$a_0m^n+a_1m^{n-1}+...+a_n=0$$ as the root(zeroes) of the auxiliary equation!

My question is

1)Why we must have have the real coefficients?

2)When does the theorem become invalid?

3)I notice that we frequently use it when the solution to the differential equation is not given.

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  1. It is not necessary, complex coefficients are well possible. But only with real coefficients will you always get a real solution space.

  2. The coefficients have to be constant for this method to be valid. Review the consequences of multiple roots.

  3. Unclear what the question is.

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The Theorem (due to Lagrange), does not require that the coefficients are real. They could be complex as well, and the Theorem still holds.

However, not all the solutions are of the form $\mathrm{e}^{mt}$. For example, in the case of $$ x''-2x'+x=0, $$ Both, $\mathrm{e}^{t}$ and $t\mathrm{e}^{t}$ are solutions.

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