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 :)

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- Dec 27th 2011, 03:32 AMmaths13first 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 AMAckbeetRe: first order homogenous equations
Either $\displaystyle y=ux,$ with $\displaystyle dy=u\,dx+x\,du,$ or $\displaystyle x=vy,$ with $\displaystyle dx=v\,dy+y\,dv,$ 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 $\displaystyle x=vy$ whenever the function $\displaystyle M(x,y)$ is simpler than $\displaystyle N(x,y)$.

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

is the homogeneous DE under discussion.

Does that answer your question? - Dec 27th 2011, 05:10 AMProve ItRe: first order homogenous equations
Yes there are others. The only way to know which to use is experience.

- Dec 27th 2011, 07:02 AMmaths13Re: 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 AMAckbeetRe: first order homogenous equations
- Jan 3rd 2012, 07:56 AMHallsofIvyRe: 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!