When Can You Divide a Differential Equation?

[inglezakis]

The point of my question is that when we divide a differential equation by a function or variable we result in different solution (not always). Take the example:

ydx+ydy=0, constaint: xy=a

By substituting x with y/a and after some manipulations we arrive to

(-a/y)dy+ydy=0 and on integration we have -aln(y2/y1)+(y2^2-y1^2)/2=0

Suppose we divide the original equation by y. We have

dx+dy=0 and then the solution is (x2-x1) + (y2-y1) =0 (the constraint is eliminated by eliminating y)

Or, we divide by (y^2), we have

dx/y+dy/y=0 and the solution is (x2^2-x1^2)/2a+ln(y2/y1)=0

Here the constraint is needed in order to substitute y for x in the dx term.

So, by dividing the original equation we arrive in different solutions. I don't have the space here, but it can be shown that the 3 solutions above converge for small (x1-x2). Anyway, the issue is when and under what condition we are allowed to divide the differential equation, the purpose being to arrive in the same solution.

 

[jvicens]

I appreciate your question and the thought process behind it. It's true that dividing a differential equation by a function or variable can result in different solutions, and it's important to understand when and under what conditions we can do so.

First, let's address why dividing by a function or variable can result in different solutions. This is because when we divide, we are essentially changing the form of the equation and therefore changing the mathematical operations needed to solve it. In your examples, dividing by y or y^2 changes the form of the equation and therefore results in different solutions.

Now, onto the question of when we can divide a differential equation. The short answer is that it depends on the specific equation and the conditions given. In some cases, dividing by a function or variable may be allowed and in others it may not be. It really depends on the mathematical properties of the equation and the constraints given.

In general, it is always important to pay attention to the constraints given in a differential equation. These constraints are often there for a reason and can impact the validity of any manipulations we make to the equation.

In your first example, the constraint xy=a is crucial for the substitution you made to work. Without this constraint, dividing by y would not be allowed and would result in an incorrect solution. In your second example, dividing by y^2 is allowed because the constraint xy=a is still present and allows for the substitution of y for x.

In conclusion, the key takeaway is to always pay attention to the constraints given in a differential equation and to understand the mathematical properties of the equation before making any manipulations. Dividing by a function or variable can result in different solutions, so it's important to use caution and ensure that any manipulations we make are valid.