Homogeneous Differential Equations.

**Showcase_22** · October 7th 2008, 02:06 PM

Originally Posted by Chris L T521

Here is one way to test and see if its homogeneous:

We (in a sense) see that $N(x,y)=\frac{1-3y}{y}$ and $M(x,y)=\frac{2-3x}{x}$ after getting our equation in the form of $M(x,y)\,dx+N(x,y)\,dy=0$

I see where this is bit is from, so great!

It is homogeneous if we can observe the following:

$M(tx,ty)=t^{\alpha}M(x,y)$ AND $N(tx,ty)=t^{\alpha}N(x,y)$ .

For this part, what are tx and ty? It's just that the DE in question is using y's and x's.

Is t a dummy variable or something?

**Chris L T521** · October 7th 2008, 02:19 PM

Originally Posted by Showcase_22

I see where this is bit is from, so great!

For this part, what are tx and ty? It's just that the DE in question is using y's and x's.

Is t a dummy variable or something?

Sorta. The variable t is like a dummy variable, but its something that can help us show [analytically] if the functions M(x,y) and N(x,y) [from the DE] in question are homogeneous.

If we introduce this dummy variable t and show that if we have $t^{\alpha}M(x,y)$ after manipulating $M(tx,ty)$ we're almost done showing that its homogeneous. The last part is to show that we get $t^{\alpha}N(x,y)$ after manipulating $N(tx,ty)$ AND that the we have the same $\alpha$ exponents. If the exponents are different, OR we can't show that $M(tx,ty)=t^{\alpha}M(x,y)$ or $N(tx,ty)=t^{\alpha}N(x,y)$ , then its not homogeneous.

Does this help?

--Chris

**Showcase_22** · October 8th 2008, 12:42 AM

I think i'm understanding it, but there's just something else that's puzzling.

We have:

$-\frac{2-3x}{x}dx+\frac{3y-1}{y}dy=0$

and we're comparing this with:

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

Comparing coefficients we get:

$M(x,y)=-\frac{(2-3x)}{x}$ and $N(x,y)=\frac{3y-1}{y}$

Is this equal to:

$M(x)=-\frac{(2-3x)}{x}$ and $N(y)=\frac{3y-1}{y}$ ?

For the next part:

$<br /> M(tx,ty)=t^{\alpha}M(x,y)<br />$

$\frac{-(2-3tx)}{tx}=t^\alpha \frac{-(2-3x)}{x}$

This gives:

$\frac{-(2-3tx)}{t}=t^\alpha (3x-2)$

$3tx-2=t^{\alpha +1} (3x-2)$

and since the -2 on the LHS does not contain a t then there is no value of $\alpha$ for which this is true.

I haven't tried it with the y's yet since it has to satisfy both these conditions to be homogeneous. Since this one doesn't work the DE is non-homogeneous.

Is that about right?

**Chris L T521** · October 8th 2008, 08:42 AM

Originally Posted by Showcase_22

I think i'm understanding it, but there's just something else that's puzzling.

We have:

$-\frac{2-3x}{x}dx+\frac{3y-1}{y}dy=0$

and we're comparing this with:

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

Comparing coefficients we get:

$M(x,y)=-\frac{(2-3x)}{x}$ and $N(x,y)=\frac{3y-1}{y}$

Is this equal to:

$M(x)=-\frac{(2-3x)}{x}$ and $N(y)=\frac{3y-1}{y}$ ?

For the next part:

$<br /> M(tx,ty)=t^{\alpha}M(x,y)<br />$

$\frac{-(2-3tx)}{tx}=t^\alpha \frac{-(2-3x)}{x}$

This gives:

$\frac{-(2-3tx)}{t}=t^\alpha (3x-2)$

$3tx-2=t^{\alpha +1} (3x-2)$

and since the -2 on the LHS does not contain a t then there is no value of $\alpha$ for which this is true. Correct

I haven't tried it with the y's yet since it has to satisfy both these conditions to be homogeneous. Since this one doesn't work the DE is non-homogeneous. Correct. You will end up with the same situation with the y's, where there is no $\alpha$ value where this would be true.

Is that about right?

It seems like you have the hang of this.

--Chris

**Showcase_22** · October 8th 2008, 01:31 PM

Sweet!

Thanks for your help Chris!

It was also nice to know that my 200th post was such a success! 8-)

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