# ODE's

Discussion in 'Math' started by Hello, Apr 16, 2009.

1. ### Hello Thread Starter Active Member

Dec 18, 2008
82
0
Using the substitution y = 1/z, find the solution of the following differential equation:

dy/dx = y + x(y^2)

This is what i've done so far:

dy/dx = (dy/dz).(dz/dx)

Not sure what to do next or if it is correct!

Any help would be greatly appreciated.

Last edited: Apr 16, 2009
2. ### mattc82 Member

Mar 13, 2009
22
0
I'm not 100% but I would seperate and integrate in y and x first,then sub y=1/z after. Unless this is supposed to be PDE?

3. ### steveb Senior Member

Jul 3, 2008
2,432
469
That is a good start. Now complete the substitution in the entire equation. The dy/dz can be easily identified from y=1/z (i.e. dy/dz=-1/z^2)

Once you make the substitution you will see that the final equation is in a very simple form.

dz/dx=-z-x

This is a simple first order system with a negative ramp (-x) as the input function.

4. ### Ratch New Member

Mar 20, 2007
1,068
4
Hello,

dy/dx = (-1/z^2)dz/dx

so (-1/z^2)dz/dx = 1/z + x/z^2 ==> -dz/dx = z + x ==> -dz = zdx + xdx

dz + zdx = -xdx ==> (e^x)dz + z(e^x)dx = -x(e^x)dx ==> d((e^x)z) = d((e^x-x(e^x)) ==> (e^x)z = (e^x-x(e^x) +C ==> z = 1-x+C/e^x
==> 1/y = 1-x+C/e^x ==> y = 1/(1-x+C/e^x)

Ratch