homogeneous differential equation problem

[PG1995]

Hi

Please have a look on the attachment. It has my queries there. Please help me with them. It would be really kind of you. Thanks.

Regards
PG

 

[MrAl]

Hi,


For the first part of your Q1 question here...

Ask yourself this, how did (x^2+u^2*x^2)*dx factor into x^2*(1+u)*dx ? How did u^2 turn into just plain u ?
Answer: it doesnt factor like that. The ENTIRE line has to be factored in one shot, not in parts. So he didnt really 'replace' anything, he just factored the entire expression while it appeared at first glance that it could be factored piece by piece.

Try it again, only this time, expand the ENTIRE expression first before you try to factor anything. This means multiply everything out and then get all the terms separate without any parens. You'll see the solution that way.

The expression we are talking about here is the one just to the left where you typed "Q1" in red. Expand that out completely first, then factor.

For the second part, Q2, you can take the expression just above the orange highlight and add y/x to both sides, then subtract ln(c) from both sides to start with. Then you can write:
ln(1+y/x)
as
ln(x/x+y/x) or ln((x+y)/x)

(using absolute value signs)

then you can combine the '2' into the ln((x+y)/x) part, then combine the ln(c) into that part too by the properties of logs.

You should really take a quick look at properties of logs on the web to get some idea what is going on here.
For one example, we have this identity:
log(a/b)=log(a)-log(b)
and obviously you can do:
log(a)-log(b)=log(a/b)