Hello,

I have the equation

, where t>0, and I know that

.

I would then like to use reduction of order to find some v(t), but I am getting a bit stuck, so I will just show my workings and hopefully someone will pick up on what I am doing wrong, as I remain with not only the derivatives of v(t) but v(t) itself, and am unsure of how to remove it so I can proceed.

v(t) is the function I'd like to find with my current known solution,

. I then found the derivatives of all the

's, plugged in, but I could not proceed as I still had the v(t) term whereas I'm supposed to be only left with its derivatives.

Plugging in:

And this leads me to believe that the terms

and

are the ones that are 'supposed' to cancel out and that I'm adding one too many t' to the equation, but I don't really see how that's wrong 'cause it's what I get when I repeat it, as well.

Hopefully someone will spot it, thanks!