1. 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

2. Re: 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:

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 $\displaystyle N(x,y)$.
Note that in Zill's notation, the DE

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

is the homogeneous DE under discussion.

3. Re: first order homogenous equations

Yes there are others. The only way to know which to use is experience.

4. 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

5. Re: first order homogenous equations

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?

6. 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!