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.

Yes, you are correct in your understanding of finding eigenfunctions of operators. To determine if a function is an eigenfunction of an operator, you apply the operator to the function and see if the result is the original function multiplied by some constant.

Let's start by finding the first derivative of the function sin(x)*e^(ax), where a is a constant.

Using the product rule of differentiation, the first derivative is given by:

d/dx(sin(x)*e^(ax)) = (cos(x)*e^(ax)) + (sin(x)*ae^(ax))

Now, we can check if the derivative is proportional to the original function. From the equation above, if we divide the derivative by the original function, we should get a constant:

[(cos(x)*e^(ax)) + (sin(x)*ae^(ax))] / (sin(x)*e^(ax)) = (cos(x)*e^(ax))/(sin(x)*e^(ax)) + a

Simplifying further, we have:

(cos(x)*e^(ax))/(sin(x)*e^(ax)) + a = cot(x) + a

Since cot(x) + a is not a constant, the original function sin(x)*e^(ax) is not an eigenfunction of the operator d/dx.

Now, let's check if the function is an eigenfunction of the operator d^2/dx^2 (the second derivative). Taking the second derivative of the original function, we have:

d^2/dx^2(sin(x)*e^(ax)) = -sin(x)*e^(ax) + 2a*cos(x)*e^(ax) - a^2*sin(x)*e^(ax)

Again, let's divide this by the original function:

[-sin(x)*e^(ax) + 2a*cos(x)*e^(ax) - a^2*sin(x)*e^(ax)] / (sin(x)*e^(ax))

Simplifying further, we have:

-sin(x)*e^(ax)/(sin(x)*e^(ax)) + 2a*cos(x)*e^(ax)/(sin(x)*e^(ax)) - a^2*sin(x)*e^(ax)/(sin(x)*e^(ax))

This simplifies to:

-1 + 2a*cos(x) - a^2*sin(x)

Since -1 + 2a*cos(x) - a^2*sin(x) is not a constant, the original function sin(x)*e^(ax) is also not an eigenfunction of the operator d^2/dx^2.

Therefore, the function sin(x)*e^(ax) with a constant 'a' is not an eigenfunction of the operators d/dx and d^2/dx^2.