# Solving an exact differential equation

• Jimmy25
In summary, the partial derivatives of M(x,y) and N(x,y) that I end up with after multiplying by the integration factor are not equal.
Jimmy25
My work is an attached photo.

I'm trying to solve this differential equation by making it exact (finding an integrating factor etc.) I've been looking at this problem for a few hours now and I can't figure out what it is that I'm doing wrong! The partial derivatives of M(x,y) and N(x,y) that I end up with after multiplying by the integration factor are not equal. Can anyone spot my error?

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• photo.jpg
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How about posting your work so that we can see it without having to tilt our heads 90 degrees?

Mark44 said:
How about posting your work so that we can see it without having to tilt our heads 90 degrees?

This should work.

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• PDF.pdf
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Your integrating factor calculation is correct. When I differentiated your expression (including the integrating factors) I get them to be the same thing. Your calculation of the final differentials are incorrect...at least to me anyhow.

rock.freak667 said:
Your integrating factor calculation is correct. When I differentiated your expression (including the integrating factors) I get them to be the same thing. Your calculation of the final differentials are incorrect...at least to me anyhow.

I don't see how they could possible be the same. When you differentiate M with respect to y (using the product rule) you get a y2 in your answer. However when you differentiate N with respect to x you get only a y and not a y2 in your answer. I've used my symbolic calculator to check my differentiation also and I get the same answer.

I still can't see what it is that I'm screwing up here!

anyone?

Your M in the denominator testing for mu is missing a factor of y. You should come up with an integrating factor of ey.

Thank you! I knew it must have been something stupid that I wasn't seeing.

## 1. What is an exact differential equation?

An exact differential equation is a type of ordinary differential equation where the solution can be found by taking the partial derivatives of a function. This means that the equation can be solved without any guesswork or trial-and-error methods.

## 2. How do you determine if a differential equation is exact?

A differential equation is exact if the coefficients of the variables and their partial derivatives are in the same order. This means that the equation can be rearranged to have the form of M(x,y)dx + N(x,y)dy = 0, where M and N are functions of x and y.

## 3. What is the process for solving an exact differential equation?

The process for solving an exact differential equation involves finding a function u(x,y) that satisfies the equation M(x,y)dx + N(x,y)dy = 0. This can be done by taking the partial derivative of u with respect to x and setting it equal to M(x,y), and then taking the partial derivative with respect to y and setting it equal to N(x,y). Once the function u is found, the solution to the original differential equation can be written as u(x,y) = c, where c is a constant.

## 4. Are there any special cases when solving an exact differential equation?

Yes, there are two special cases. The first case is when M(x,y) = N(x,y), which means the equation is already exact and the solution can be found by simply integrating both sides. The second case is when M(x,y) = -N(x,y), which means the equation is not exact but can be made exact by multiplying the entire equation by a function of x and y.

## 5. How can solving an exact differential equation be applied in real-world situations?

Exact differential equations are commonly used in physics and engineering to model various systems, such as fluid flow, heat transfer, and electrical circuits. They can also be used to solve optimization problems and in economics to model supply and demand functions.

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