First order non linear differential equation

[Eide]

I have tryed every method I can think of, and Im not getting any results of solving this one... If anyone can help me I would be very happy!

dx/dt = (t^2+3tx+x^2)/(t^2)

Here is how the answear should be:
http://www.wolframalpha.com/input/?i=dx/dt+=+(t^2+3tx+x^2)/(t^2)

 

[Ishuda]

I have tryed every method I can think of, and Im not getting any results of solving this one... If anyone can help me I would be very happy!

dx/dt = (t^2+3tx+x^2)/(t^2)

Here is how the answear should be:
http://www.wolframalpha.com/input/?i=dx/dt+=+(t^2+3tx+x^2)/(t^2)

Off the top of my head, there may be an easier way (integration by parts?) but I know that if I have a problem with a dx/dt=x/t+... hanging around a place to start looking may be x=tf(t) where f is some function, i.e.
$\displaystyle \frac{dx}{dt} = f(t) + t \frac{df}{dt} = 1 + \frac{3x}{t} + \frac{x^2}{t^2}= 1 + 3 f(t) + f^2(t)$

and that, IMO, is an easier problem to handle.

 

[HallsofIvy]

If you do the "indicated" division on the right, you get $\displaystyle \frac{dx}{dt}= 1+ 3\frac{x}{t}+ \left(\frac{x}{t}\right)^2$ and the substitution $\displaystyle y= \frac{x}{t}$ seems obvious. $\displaystyle x= ty$ so that $\displaystyle \frac{dx}{dt}= t\frac{dy}{dt}+ y$. The equation becomes $\displaystyle t\frac{dy}{dt}+ y= 1+ 3y+ y^2$ which is separable.

(The key was that every term on the right was of degree 2.)