# Simplifying non-linear ODE

• Feb 14th 2013, 05:19 AM
acid_rainbow
Simplifying non-linear ODE
Hi all,

So I have the equation:

dx/dt = (x2+xt)/t2 where x(1) = 2

I am aware that I can make a substitution by letting y equation a function of x and t, however I am unsure as to what I should let y equal and how I would plug it in and end up solving the equation.

Any help is much appreciated!

Thanks :)
• Feb 14th 2013, 06:00 AM
a tutor
Re: Simplifying non-linear ODE
$\displaystyle \frac{dx}{dt}=\left(\frac{x}{t}\right)^2+\left( \frac{x}{t}\right)$

Does that suggest anything? ;)
• Feb 14th 2013, 06:05 AM
acid_rainbow
Re: Simplifying non-linear ODE
Ya I got that! So then I get F(y) = ysq + y

Not sure how to solve this though :S
• Feb 14th 2013, 06:08 AM
a tutor
Re: Simplifying non-linear ODE
Use y=x/t (so that x=ty) and differentiate with respect to t.

dx/dt = .........
• Feb 14th 2013, 08:13 AM
acid_rainbow
Re: Simplifying non-linear ODE
okay I think I got it, this is just taking a while to click, I just need to remember you can treat most of it like normal algebra.

Did you get the constant = -0.5?

- Upon substitution I get: 1/y^2 dy = 1/t dt

chugging through and putting in initial conditions I get:

- Ln(t) - 1/y = A

Thanks again, really appreciated.
• Feb 14th 2013, 08:49 AM
a tutor
Re: Simplifying non-linear ODE
I think you have it.

In case I mislay my bit of paper again, the solution I got was $\displaystyle x=\frac{2t}{1-2\ln 2}$.

and you're welcome by the way. :) I'm relearning differential equations myself at the moment.