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we have to solve the following problem for our ODE class.

## Homework Statement

Find the solution of the initial value problem

dx/dt = (x^2 + t*x - t^2)/t^2

with t≠0 , x(t_0) = x_0

Describe the (maximal) domain of definition of the solution.

## The Attempt at a Solution

Well, I know that this is a 1st order nonlinear ODE. Unfortunately I got no clue how to deal them.

I tried this:

dx/dt = (x^2 + t*x - t^2)/t^2

= x^2/t^2 + x/t -1

Now substitute: u = x/t -> x=ut , x'=u't+u

Therefore we get:

u't+u = u^2+u-1

t* du/dt +u = u^2+u-1 //-u

t* du/dt = u^2 -1

0= t*u' -u^2 +1

which is my dead end.

Is the idea ok? What could I do?

Kind regards,

mihyaeru

PS: How can i insert a fraction?