# ?Test for Exactness of Separable Differential Equations

• Naeem
In summary, a separable differential equation is an ordinary differential equation where the variables can be separated and expressed as a product of two functions. To test for exactness, the method of partial derivatives can be used. If the equation is exact, it can be written as the total differential of a function, making it easier to solve. If the equation is not exact, it can often be made exact by multiplying by an integrating factor. These types of equations are important in science as they are used to model physical phenomena and have applications in various fields.
Naeem
Q. Prove that a separable differential equation must be exact.

Well, don't know no how to do this. There is no proof given in the textbook.

All I know,

Mdx = Ndy ( Test for exactness )

Anybody here, any ideas

separable equation has the form

$$\frac{y^\prime}{f(y)} = g(x) \Longleftrightarrow g(x) - \frac{y^\prime}{f(y)} = 0$$

apply exactness test...

The exactness of a differential equation can be determined by checking if the partial derivatives of both sides are equal. In the case of a separable differential equation, the equation can be written in the form of Mdx = Ndy, where M and N are functions of x and y.

To prove that a separable differential equation must be exact, we can use the following steps:

1. Rewrite the equation in the form of Mdx = Ndy, where M and N are functions of x and y.
2. Take the partial derivative of M with respect to y and the partial derivative of N with respect to x.
3. If the partial derivatives are equal, then the differential equation is exact. This is because the equality of the partial derivatives implies that the equation satisfies the condition for exactness, which is dM/dy = dN/dx.
4. If the partial derivatives are not equal, then the equation is not exact and cannot be solved using the method of separation of variables.

Therefore, a separable differential equation must be exact in order for it to be solved using the method of separation of variables. This is because the condition for exactness is necessary for the equation to be solvable by this method.

## What is a separable differential equation?

A separable differential equation is a type of ordinary differential equation where the variables can be separated and expressed as a product of two functions, one with respect to the dependent variable and the other with respect to the independent variable.

## How do you test for exactness of a separable differential equation?

To test for exactness, you can use the method of partial derivatives. Take the partial derivative of the function with respect to the independent variable, and then with respect to the dependent variable. If the resulting expressions are equal, the equation is exact.

## What does it mean if a separable differential equation is exact?

If a separable differential equation is exact, it means that the equation can be written as the total differential of some function. This makes it easier to solve, as the solution can be found by integrating the function.

## What if a separable differential equation is not exact?

If a separable differential equation is not exact, it can often be made exact by multiplying by an integrating factor. This factor is found by taking the ratio of the partial derivatives from the test for exactness.

## Why are separable differential equations important in science?

Separable differential equations are important in science because they are used to model many physical phenomena, such as population growth, chemical reactions, and electrical circuits. They also have applications in fields such as engineering, physics, and economics.

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