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.

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?

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.

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