The DE is of the 'Euler's type' and its solutions are of the form ... imposing that You find a second order algebraic equation in one solution of which is ... what is the other solution?...
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.
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!
You'll notice a few differences between this result and yours. In particular, there is no because I'm taking the derivative with respect to
Plugging into the DE yields
And now you can reduce the order, cancel some 's, and proceed on your merry way. Alternatively, you could notice that, right now, the LHS of the above equation is a perfect derivative, allowing you to simply integrate almost immediately. That is, you have
Take your pick.