Non exact differential equation

MHB
Thread starter danny12345
Start date Oct 12, 2015
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Differential Differential equation

In summary, a non exact differential equation is a type of differential equation that cannot be solved using standard methods because it does not satisfy the condition of exactness. To determine if an equation is non exact, one can check if the partial derivatives of the coefficients are equal. Some techniques for solving non exact differential equations include using integrating factors, power series, separation of variables, and substitution. These equations cannot be solved analytically in most cases, but can have real-world applications in fields such as physics, engineering, and biology.

Oct 12, 2015

danny12345

$y(x^3e^{xy}-y) \, dx+x(xy+x^3e^{xy}) \, dy=0$
change it into exact differential and help me in solving it

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Oct 12, 2015

Ackbach

Gold Member

MHB

I've improved your typesetting by including dollar signs, as well as introducing a right parenthesis that seemed implied.

So, we need to convert this into an exact equation. I would recommend multiplying by something of the form $x^n y^m$, and see if you can figure out the $n$ and $m$ that make it exact.

Related to Non exact differential equation

1. What is a non exact differential equation?

A non exact differential equation is a type of differential equation that cannot be solved using standard methods. This is because the equation does not satisfy the condition of exactness, which means that the partial derivatives of the equation's coefficients with respect to the variables are not equal. Non exact differential equations require special techniques or approximations to find a solution.

2. How can I tell if a differential equation is non exact?

To determine if a differential equation is non exact, you can check if the equation satisfies the condition of exactness. This means that the partial derivatives of the equation's coefficients with respect to the variables must be equal. If they are not equal, then the equation is non exact. Additionally, if a standard method of solving the equation does not work, it is likely that the equation is non exact.

3. What are some techniques for solving non exact differential equations?

Some techniques for solving non exact differential equations include using integrating factors, which involve multiplying the equation by a suitable function to make it exact. Another technique is using power series to approximate a solution. Separation of variables and substitution are also commonly used methods for solving non exact differential equations.

4. Can non exact differential equations be solved analytically?

In most cases, non exact differential equations cannot be solved analytically, meaning a closed-form solution cannot be found. This is because the equation does not satisfy the condition of exactness, which is necessary for standard methods of solving differential equations to work. However, in some cases, special techniques or approximations can be used to find an analytical solution.

5. What are some real-world applications of non exact differential equations?

Non exact differential equations have numerous applications in fields such as physics, engineering, and biology. They are commonly used to model physical systems that involve non-linear relationships, such as fluid flow, chemical reactions, and population dynamics. They are also useful for predicting the behavior of complex systems, such as weather patterns or economic markets.