Hi
I am having troubles with proving that y=3t+t^2 is one of the solutions of ty'-y=t^2. Can someone show me? I already solved the equation without the book's help but I don't know what to do with it
I'm confused about something. You say you solved the differential equation, but you don't know if the given solution is one of them? I would think you'd simply be able to compare them.
Anyway, proving the given solution solves the differential equation is easy: Simply stick the solution into the equation:
$\displaystyle ty^{\prime} - y = t^2$
and see if it is an identity.
$\displaystyle y = 3t + t^2$
$\displaystyle y^{\prime} = 3 + 2t$
So
$\displaystyle t(3 + 2t) - (3t + t^2)$
$\displaystyle = 3t + 2t^2 - 3t - t^2$
$\displaystyle = t^2$
as desired.
-Dan
ty'-y+t^2
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.
So,
(ye^-t)'= te^-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
to apply the integrating factor method, it's best, if not required, to have the coefficient of y' to be 1. so we would divide through by t before doing anything.
you should have noticed what you had does not work. since if we multiply through by $\displaystyle e^{-t}$ we get:
$\displaystyle te^{-t}y' - e^{-t}y = t^2e^{-t}$
clearly the left hand side is not the derivative of something using the product rule. the $\displaystyle te^{-t}$ would ensure that the derivative $\displaystyle te^{-t}y$ has three terms when fully expanded.
however, if we divide by t first, we have $\displaystyle y' - \frac 1ty = t$
then $\displaystyle \mu (t) = exp \left( - \int \frac 1t~dt \right) = \frac 1t$, multiplying through we get:
$\displaystyle \frac 1ty' - \frac 1{t^2}y = 1$
and now the left side is the derivative given by the product rule of $\displaystyle \frac 1ty$
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.
i'm not sure what's your hangup. just put it in your mind that the ideal form to have is $\displaystyle y' + p(t)y = g(t)$, here $\displaystyle exp \left( \int p(t)~dt \right)$ is the integrating factor, and you are to multiply through the whole equation by it.
see post #21 here