Separable differential equation

• Syrus
In summary: This means that y = 0 does satisfy the diff. equation, but only for x = -2.In summary, the conversation is about finding a singular solution of the differential equation dy/dx = (xy+2y-x-2)/(xy-3y+x-3) and the struggle to find the best technique of integration. The suggested techniques are polynomial long division or adding and subtracting the same quantity to simplify the expressions. It is also discussed whether y=1 or y=0 is a singular solution, with the conclusion that y=0 is not a solution.
Syrus

Homework Statement

I am asked to find a singular solution of the D.E. dy/dx = (xy+2y-x-2)/(xy-3y+x-3). I am first solving to find the general solution form of the D.E., and so far have it to:

[(x+2)/(x-3)]dx = [(y+1)/(y-1)]dy

From here, of course, you integrate both sides, but I am struggilng to find the best technique of integration. Any ideas?

Syrus said:

Homework Statement

I am asked to find a singular solution of the D.E. dy/dx = (xy+2y-x-2)/(xy-3y+x-3). I am first solving to find the general solution form of the D.E., and so far have it to:

[(x+2)/(x-3)]dx = [(y+1)/(y-1)]dy

From here, of course, you integrate both sides, but I am struggilng to find the best technique of integration. Any ideas?

The Attempt at a Solution

You can use polynomial long division on each of your two rational expressions, or you can add and subtract the same quantity in each numerator so as to get expressions that are easier to work with. Both techniques produce the same results.

For example, (x + 2)/(x - 3) = (x - 3 + 5)/(x - 3) = (x - 3)/(x - 3) + 5/(x - 3) = 1 + 5/(x - 3). The same sort of idea works with the other rational expression.

Thanks Mark. That helped. Can anyone also verify that y=1 is a singular solution to the D.E.?

I have solution, which I disagree with, claiming y=0 is a singular solution since, upon substitution, it doesn't seem to produce an identity.

Syrus said:
I have solution, which I disagree with, claiming y=0 is a singular solution since, upon substitution, it doesn't seem to produce an identity.
Which should suggest that y = 0 isn't a solution at all.

If y ##\equiv## 0 is a (purported) solution, then it follows that dy/dx ##\equiv## 0. However, if y = 0, from the diff. equation, we have dy/dx = (-x - 2)/(x - 3), which is zero only if x = -2.

1. What is a separable differential equation?

A separable differential equation is a type of differential equation where the variables can be separated into two sides of the equation. This allows for the solution to be written as a product of two functions, one of which only contains the independent variable and the other only contains the dependent variable.

2. How do you solve a separable differential equation?

To solve a separable differential equation, you first need to separate the variables on each side of the equation. Then, you can integrate both sides with respect to their respective variables. Finally, you can solve for the constant of integration and obtain the general solution.

3. What is the difference between a separable differential equation and a non-separable differential equation?

A separable differential equation can be separated into two sides with only one variable on each side, while a non-separable differential equation cannot be separated in this way. This means that the solution to a separable differential equation can be written as a product of two functions, while the solution to a non-separable differential equation cannot be expressed in this form.

4. Can separable differential equations be solved using numerical methods?

Yes, separable differential equations can be solved using numerical methods such as Euler's method or the Runge-Kutta method. However, these methods are more commonly used for non-separable differential equations that cannot be solved analytically.

5. What are some real-world applications of separable differential equations?

Separable differential equations are used in various fields of science and engineering, such as physics, chemistry, and economics. They can be used to model population growth, chemical reactions, and electrical circuits, among others. They are also commonly used in mathematical modeling and data analysis.

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