Determine if the function sin(x)*e^(ax) where a=constant is an eigenfunction of the operators d/dx and d^2/(dx)^2
Okay. My understanding is that you use the operator and perform its "thing" on the function. In this case, you will have to find the 1st derivative of sin(x)*e^(ax) ... And if the result is sin(x)*e^(ax) multiplied by some constant, it is an eigenfunction.
Is this correct? I am doing the first derivative but it doesn't show up as an eigenvalue. Neither does the 2nd derivative :(
You are absolutely right. To be an eigenfunction, the operator has to reproduce the function with some multiplicative constant. Even without doing a lot of work, you can see for the special case of a=0 it doesn't work because you need to take four derivatives of sine to get back to sine. The problem says "determine if" so you can just say no. :)
Thank you! I thought I was going crazy because I wasn't able to an eigenfunction since 1st deriv. would be
= cos(x)*e^(ax) + a*sin(x)*e^(ax)
which is definitely not an eigenfunction of the operator.