Anyway, proving the given solution solves the differential equation is easy: Simply stick the solution into the equation:
and see if it is an identity.
Ok, this is my first time dealing with equations like these so I might be wrong. Here's how I did it. (A little extra info about my background: I've only been exposed to these type of problems for 2 1/2 days)
The integrating factor here is e^-t since p(t)= -1 and integrating you would get -t.
u=t du= dx dv=e^-t dx
Integrating both sides, I finally get
y= -t- 1 +Ce^t
Please tell me if this is wrong when you see it
you should have noticed what you had does not work. since if we multiply through by we get:
clearly the left hand side is not the derivative of something using the product rule. the would ensure that the derivative has three terms when fully expanded.
however, if we divide by t first, we have
then , multiplying through we get:
and now the left side is the derivative given by the product rule of
and we're in business
I understand now how you got that but I'm a little shaky with these IF's because I tend to look at them by using the formula e^integral p(t) dx. I did divide by t on both sides before I multiplied the IF on both sides.
Thanks for the help. That is certainly something I've never seen before.