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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
Any ideas?
Thanks in advance!
-epiclesis
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.
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.
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.
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.
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.