# Diff. Equation's General Formula use Substitution

• Nov 1st 2006, 12:28 AM
asteg123
Diff. Equation's General Formula use Substitution
I'm really stomped with this problem... i can't seem to get the answer...

anyway... here's the problem..

(2(x^3) - (y^3))y'=3(x^2)y

and i need to get the general solution....

SO, here's what i did...

I Let
u=x^3 and
du=3x^2dx

so what happens is

2udy-(y^3)dy= ydu

and that's where i got stuck...

i tried using
d(u/y)=(ydu-udy)/y^2

but i can't seem to get rid of the 2 in 2udy and it would be much of a problem if i did that...

could anyone help me??
• Nov 1st 2006, 07:45 AM
ThePerfectHacker
Quote:

Originally Posted by asteg123
I'm really stomped with this problem... i can't seem to get the answer...

anyway... here's the problem..

(2(x^3) - (y^3))y'=3(x^2)y

What we have is,
$\displaystyle y'=\frac{3x^2y}{2x^3-y^3}$
This is a homogenous equation which can be transformed into a seperate equation.
If you use the substitution,
$\displaystyle y=vx$<---->$\displaystyle y/x=v$
Then,
$\displaystyle y'=v+v'x$
If you divide by the numerator and denominator you have,
$\displaystyle y'=\frac{3}{2(x/y)-(y/x)^2}$
Thus,
$\displaystyle v+v'x=\frac{3}{\frac{2}{v}-v^2}$