Solve For Exact Differential Equation

In summary, the conversation discusses solving the exact equation y^3-(14x+2)dx+3xy^2dy=0 by first proving that it is exact and then finding the solution. The solution involves working with the function N and taking its derivative, which leads to finding the value of h(x). Plugging this into the original equation results in f(x,y)=xy^3-7x^2+2x=C, which can be solved for y to obtain the correct solution. However, the original attempt had a basic algebra error that was corrected.
  • #1
TrueStar
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Homework Statement



Solve the exact equation y^3-(14x+2)dx+3xy^2dy=0

Homework Equations



NA

The Attempt at a Solution



I proved these were exact because dM/dy and DN/dx both equal 3y^2

I chose to work with N first and df/dy=3xy^2

Therefore f(x,y)=xy^3+h(x)

I took df/dx of this and got y^3 + h'(x)

I made this equal to the other df/dx so it looks like:
df/dx = y^3+h'(x)=y^3-(14x+2)

h'(x)=-14x+2 and so h(x) is -7x^2+2x (+ constant)

I plugged this into the original problem with h(x) so it now looks like:

f(x,y)=xy^3-7x^2+2x=C

Solving for y, I get y=(-7x^2+2x+C)^(1/3) / x^(1/3)

This is not correct and I don't know what I'm missing here.
 
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  • #2
You have a basic algebra error! -(14x+ 2)= -14x- 2, not -14x+ 2.
 
  • #3
Ah yes. I did it again from the beginning and was able to solve it correctly. Thanks for pointing out the error. :)
 

FAQ: Solve For Exact Differential Equation

What is an exact differential equation?

An exact differential equation is a type of differential equation where the independent and dependent variables can be interchanged without changing the equation's form. This means that the equation can be solved by integrating both sides with respect to one of the variables.

How do you determine if a differential equation is exact?

A differential equation is exact if the partial derivatives of its terms with respect to the variables are equal. This means that the order of differentiation does not matter.

What is the process for solving an exact differential equation?

To solve an exact differential equation, you must first determine if it is exact by checking if the partial derivatives are equal. If it is exact, you can then integrate both sides of the equation with respect to one of the variables. This will result in a solution that includes an arbitrary constant.

Are there any special cases when solving exact differential equations?

Yes, there are some special cases that may arise when solving exact differential equations. For example, there may be a possibility of the solution being a function of both variables instead of just one. In this case, the solution will be a family of curves rather than a single curve.

What are some applications of exact differential equations in science?

Exact differential equations have various applications in science, particularly in fields such as physics, chemistry, and engineering. They can be used to model and analyze physical systems, chemical reactions, and other natural phenomena. They are also essential in solving problems related to motion, heat transfer, and fluid dynamics.

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