# First Order ODE using homogeneous substituion

• Mar 1st 2008, 04:20 PM
ramzouzy
First Order ODE using homogeneous substituion
someone please enlighten me on each step taken (as well as integrating both sides of the separable equation involving z and x) in solving the following equation:

(x^2)y'=7(y^2)-2xy
using the subsitution z=(y/x)

i know what has to be done, but im having troubles simplifying into a regular funtion y(x)...

this can be done using bernouilli's method but wanna get the same using the homogeneous substituion method

thanks
• Mar 1st 2008, 04:43 PM
Krizalid
$y' = 7 \cdot \left( {\frac{y}
{x}} \right)^2 - 2 \cdot \left( {\frac{y}
{x}} \right).$

Now make $y=xz.$
• Mar 1st 2008, 05:00 PM
ramzouzy
yea i did that, but solve down to the part where you end up with a separable equation dz/7(z^2)-3z = dx/x.

i tried integrating both sides, and especially the left one needed more tactics where i used partial integration {dz/7(z^2)-3z} = 1/7[A/z + B/{z-(3/7))]

my problem here is integrating and getting a solution y(x) after back substituting z=(y/x). I did the same ODE using bernouillis and i got
y(x)=-7x^(-1) +Cx^(-2). Just want to see how to reach the same results using the homogenous substitution technique