- #1
phoneketchup
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So I have an integral:
## \delta W=\int_{-\Delta}^\Delta\left[x^2\left(\frac{d\xi}{dx}\right)^2−D_S\xi^2\right]dx ##
Here ##\xi## is a function of ##x## and ##D_S## is a constant. ##\Delta## is just some small ##x##. Now I need to set the variation of ##\delta W## to 0. Do do this I differentiated whatever is inside the bracket and set it to 0. I get:
## x^2\xi″+x\xi′−D_S\xi = 0 ##
However, the answer is:
## \frac{d}{dx}\left(x^2\frac{d\xi}{dx}\right)+D_S\xi = x^2\xi″+2x\xi′+D_S\xi = 0 ##
Where the primes are derivatives with respect to x. As you can see the difference is a factor of 2 in the middle term and that minus sign.
If anyone could point out where I am going wrong, it would be really appreciated.
Thanks!
## \delta W=\int_{-\Delta}^\Delta\left[x^2\left(\frac{d\xi}{dx}\right)^2−D_S\xi^2\right]dx ##
Here ##\xi## is a function of ##x## and ##D_S## is a constant. ##\Delta## is just some small ##x##. Now I need to set the variation of ##\delta W## to 0. Do do this I differentiated whatever is inside the bracket and set it to 0. I get:
## x^2\xi″+x\xi′−D_S\xi = 0 ##
However, the answer is:
## \frac{d}{dx}\left(x^2\frac{d\xi}{dx}\right)+D_S\xi = x^2\xi″+2x\xi′+D_S\xi = 0 ##
Where the primes are derivatives with respect to x. As you can see the difference is a factor of 2 in the middle term and that minus sign.
If anyone could point out where I am going wrong, it would be really appreciated.
Thanks!