> In trying to utilize the method of unknown coefficients I ended up
> with this doozy:
>
> y=A*Exp(3*t)+B*t*Exp(3*t)+C*t^2*Exp(3*t)+D
>
> How can I get sage to treat A-D as arbitrary constants when
> differentiating.

You just have to tell sage to think of them as variables:

sage: var('A B C D t')
(A, B, C, D, t)

sage: y = A*exp(3*t) + B*t*exp(3*t) + C*t^2*exp(3*t) + D

sage: y.differentiate(t)
3*t^2*e^(3*t)*C + 2*t*e^(3*t)*C + 3*t*e^(3*t)*B + e^(3*t)*B + 3*e^(3*t)*A

Hope that helps ...

-cc

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