Solution to the diff. eqn. dy/y-a*dx/x = b, where a and b are constant

[lar739]

It is very clear that the solution to the equation "dy/y-a*dx/x = 0" is y=C*x^a. However I cannot figure out the solution when I add the constant b to the other side. Any help would be greatly appreciated!

 

[maajdl]

Integrate both sides of the equation dy/y - a dx/x = b .

 

[lar739]

Thanks for the hint but, integrate b with respect to what?

 

[maajdl]

Well, I assume y is a function of a parameter -say- p: y(p).
The same for x.

And I assume the right hand side must have been some b dp.
No idea what the exact question could be, of course.

If this comes from a book, you could provide the reference.

So my assumption is: you are asked to solve

dy/y - a dx/x = bdp

You need anyway a d-'something' (differential) on the rhs.
So usual, that I even did not see it was missing in your question.

 

[lar739]

The equation is not from any book, I derived it for a particular problem that I have.

y is a function of x, y(x) and a and b are two constants.

I do not know if it has a solution. The thing is that if b=0, the solution is very simple, y=C*x^a, so I was wondering whether there is a closed form solution if b is non-zero but a constant.

 

[maajdl]

The right hand side must be a differential, not a simple constant.
For example, you could rewrite the original equation as:

1/y dy/dx - a /x = 0

Here you have a derivative on the left hand side, not a differential anymore.
You can then generalize it to:

1/y dy/dx - a /x = b

And you can transform it to:

dy/y - dx/x = b dx

where "elements" or "differential" appear on both sides.
It should be like that!
Something supposed to be "as small as needed" on the left cannot be equal a given constant on the right.