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## Main Question or Discussion Point

Question:

If the roots of the characteristic equation are real, show that a solution of ay" + by' + cy = 0 is either everywhere zero or else can take on the value zero at most once.

Okay, if the roots of the CE are real, then the solution takes on one of two forms:

(i) y(t) = c_1 e^(r_1 t) + c_2 e^(r_2 t)

or

(ii) y(t) = c_1 e^(r_1 t) + c_2 t e^(r_2 t)

So now either both c_1 and c_2 are both zero, making the solution everywhere zero, or else if c_1 and c_2 are not both zero, the solution can only have the zero value at most once. I think this is because the exponential function is either always increasing or always decreasing, or because if you combine two exponential functions that it only changes from increasing to decreasing at most once (is that even correct?)......but i have no idea how to PROVE this!

maybe take critical points of the solution and show that there is at most one critical point?

can anyone help with this?

If the roots of the characteristic equation are real, show that a solution of ay" + by' + cy = 0 is either everywhere zero or else can take on the value zero at most once.

Okay, if the roots of the CE are real, then the solution takes on one of two forms:

(i) y(t) = c_1 e^(r_1 t) + c_2 e^(r_2 t)

or

(ii) y(t) = c_1 e^(r_1 t) + c_2 t e^(r_2 t)

So now either both c_1 and c_2 are both zero, making the solution everywhere zero, or else if c_1 and c_2 are not both zero, the solution can only have the zero value at most once. I think this is because the exponential function is either always increasing or always decreasing, or because if you combine two exponential functions that it only changes from increasing to decreasing at most once (is that even correct?)......but i have no idea how to PROVE this!

maybe take critical points of the solution and show that there is at most one critical point?

can anyone help with this?