Toronto Math Forum
MAT2442018F => MAT244Lectures & Home Assignments => Topic started by: Boyu Zheng on September 13, 2018, 08:08:58 PM

Hi,
I have a question from today's lecture. The initial condition equation is $y'= (xy)/(x+y2)$ and after a couple steps we get
$$dw/dt = (tw+ab)/(t+w+ab2) = tw/t+w.$$
And I am confused about why you say that $ab=0$ and $ab2=0$?

First of all, I wrote $ab=0$ and $a+b2=0$ in my notes (not $ab2=0$). And, I also wrote $x=t+a$, $y=w+b$, $x=t+1$, $y=w+1$, which may be the initial condition. In this case, $a=1$, and $b=1$, then the equation will make sense.
Well, actually I am confused about it too...

We changed $x=t+a$, $y=w+b$ with constants $a,b$ to be chosen later and got
$$
\frac{dw}{dt}=\frac{tw+ab}{t+w+ab2}
\tag{*}
$$
We want to have a homogeneous equation in $(t,w)$ and it happens if $ab=0$, $ab2=0$. So, we choose $a,b$ to satisfy these equations.

Is it because there is a redundant constant 2, so we would like to change from the equation of (x,y) to (w,t)?

Is it because there is a redundant constant $2$, so we would like to change from the equation of $(x,y)$ to $(w,t)$?
Indeed