?Test for Exactness of Separable Differential Equations

[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

 

[Data]

separable equation has the form

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

apply exactness test...

 

[M98Ranger]

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.