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

• lar739
In summary: If it were, the right hand side would be just the left hand side multiplied by the constant, and that's not the case.In summary, the equation is dy/y - a dx/x = b dx where b is a constant. The equation has a solution if b is a constant, but it is not a simple solution.
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!

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

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

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.

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.

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.

## 1. What is a "differential equation"?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is commonly used to model various physical, chemical, and biological processes.

## 2. What does "dy/y-a*dx/x = b" mean?

This is a specific form of a differential equation, where y and x are the dependent and independent variables, respectively. "dy/y" and "dx/x" represent the derivatives of y and x, respectively. The constants a and b are coefficients in the equation.

## 3. How do you solve a differential equation?

There are various methods for solving differential equations, depending on the type and complexity of the equation. In this particular case, the equation can be solved using the separation of variables method, which involves isolating the variables on opposite sides of the equation and integrating both sides.

## 4. What is the significance of the constants a and b in the equation?

The constants a and b determine the behavior of the solution to the differential equation. They can represent physical parameters, initial conditions, or other factors that affect the behavior of the system being modeled.

## 5. Can this differential equation have multiple solutions?

Yes, depending on the values of the constants a and b, there can be multiple solutions to this equation. In some cases, the solution may also be a family of curves rather than a single curve.

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