first order homogenous equations

• Dec 27th 2011, 03:32 AM
maths13
first order homogenous equations
We've just started differential equations and the subsitution we use is y=vx, I was wondering whether thats the only substitution to use or whether there are others...and if there are others, how will I know which one to use?

Thanks :)
• Dec 27th 2011, 05:04 AM
Ackbeet
Re: first order homogenous equations
Either $y=ux,$ with $dy=u\,dx+x\,du,$ or $x=vy,$ with $dx=v\,dy+y\,dv,$ will reduce a first-order homogeneous equation to a separable one. Here's a quote from Zill, 6th Ed., p. 55:

Quote:

Although either of the indicated substitutions can be used for every homogeneous differential equation, in practice we try $x=vy$ whenever the function $M(x,y)$ is simpler than $N(x,y)$.
Note that in Zill's notation, the DE

$M(x,y)\,dx+N(x,y)\,dy=0$

is the homogeneous DE under discussion.

• Dec 27th 2011, 05:10 AM
Prove It
Re: first order homogenous equations
Yes there are others. The only way to know which to use is experience.
• Dec 27th 2011, 07:02 AM
maths13
Re: first order homogenous equations
Ahh thank you...it makes a bit more sense now! So what would I use for (3x^2)y(dy/dx)= x^3 + 2y^3

Thanks again :)
• Dec 27th 2011, 07:04 AM
Ackbeet
Re: first order homogenous equations
Quote:

Originally Posted by maths13
Ahh thank you...it makes a bit more sense now! So what would I use for (3x^2)y(dy/dx)= x^3 + 2y^3

Thanks again :)

Well, what does the DE look like in the form of Post # 2?
• Jan 3rd 2012, 07:56 AM
HallsofIvy
Re: first order homogenous equations
Quite frankly, it is probably simplest just to always use y= xv rather than take the time to worry about it!