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- When solving a differential equation

Good Morning

I have a second order, linear differential equation. I solve it with complex exponentials. When I am done, I contract terms producing this:

q = exp

Then I define new constants (prior to using the initial conditions)

A = C+D

B = i(C-D) // I forgot the "i" here when I first posted this, but the rest of my post still has the question that I am asking. My dropping of the "i" here was a grave mistake in stating my own question... I am sorry.

And I have

q = exp

OK; what happened to the "i?"

Now, I know I can leave the original constants and work those through.

I also have a "feeling" that the new numbers COULD involve anything, even imaginary terms.

And I work it out. So all I did was assign B and ignored the "i"

But I always get a funny feeling at this step, and would like (hope) for someone to explain: "why can I justify, at that moment, why I can drop the "i" from consideration of the new definition B. No book I have read, ever addresses this point, and it may, indeed, be silly/Obvious, but I would really appreciate some thoughts by others.

I mean, I get taht I can say "iB is just a constant." But to my stubborn mind, it is a constant from the imaginary world.

I have a second order, linear differential equation. I solve it with complex exponentials. When I am done, I contract terms producing this:

q = exp

^{(some term regarding damping)}* ( C+D) cos(wt) + i (C-D) sin(wt) )Then I define new constants (prior to using the initial conditions)

A = C+D

B = i(C-D) // I forgot the "i" here when I first posted this, but the rest of my post still has the question that I am asking. My dropping of the "i" here was a grave mistake in stating my own question... I am sorry.

And I have

q = exp

^{(some term)}* Acos(wt) +Bsin(wt) )OK; what happened to the "i?"

Now, I know I can leave the original constants and work those through.

I also have a "feeling" that the new numbers COULD involve anything, even imaginary terms.

And I work it out. So all I did was assign B and ignored the "i"

But I always get a funny feeling at this step, and would like (hope) for someone to explain: "why can I justify, at that moment, why I can drop the "i" from consideration of the new definition B. No book I have read, ever addresses this point, and it may, indeed, be silly/Obvious, but I would really appreciate some thoughts by others.

I mean, I get taht I can say "iB is just a constant." But to my stubborn mind, it is a constant from the imaginary world.

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