All separable equations are exact?

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In summary, an exact separable equation is a type of differential equation that can be solved using the method of exactness. A separable equation is considered exact if it is written in a specific form and the partial derivatives of its functions meet certain criteria. The method of exactness involves finding an integrating factor to make the equation easier to integrate and solve. Not all separable equations are exact and may require different methods to solve. To solve an exact separable equation, one must check for exactness, find the integrating factor, and then solve for the solution. The solution can be checked by plugging it back into the original equation.
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epiclesis
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How can I show that the separable equation dy/dx = M(x)N(y) is also exact?

Any ideas?

Thanks in advance!

-epiclesis
 
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  • #2
dy/dx = M(x)N(y) implies
M(x)dx - dy/N(y) = 0
is this equation exact?
 
Last edited:
  • #3
Of course, that should be "M(x)dx- N(y)dy= 0".
 

1. What is the definition of an exact separable equation?

An exact separable equation is a type of differential equation where the independent and dependent variables can be separated and the resulting terms can be rearranged to form an exact differential equation. This means that the equation can be solved using a specific method called the method of exactness.

2. How do you know if a separable equation is exact?

A separable equation is considered exact if it meets the following criteria: the equation is written in the form of f(x,y)dx + g(x,y)dy = 0, where f and g are functions of x and y, and the partial derivative of f with respect to y is equal to the partial derivative of g with respect to x.

3. What is the method of exactness used to solve separable equations?

The method of exactness involves finding a function called the integrating factor, which can be multiplied to both sides of an exact separable equation to make it easier to integrate. This allows for the equation to be solved and the solution to be found.

4. Are all separable equations exact?

No, not all separable equations are exact. Only a specific type of separable equation that meets the criteria mentioned above can be considered exact. Other types of separable equations may require different methods or techniques to solve.

5. How do you solve an exact separable equation?

To solve an exact separable equation, you must first check if it meets the criteria for exactness. Then, you can use the method of exactness to find the integrating factor and multiply it to both sides of the equation. This will allow you to integrate and solve for the solution. Finally, you can check your solution by plugging it back into the original equation to ensure it satisfies the equation.

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