Non-exact differential equation

In summary, the problem is a non-separable, non-homogeneous and non-exact differential equation. The person tried to solve it by finding an integrating factor, but encountered difficulties due to the presence of two variables. However, by applying a change of variables, the equation can be transformed into a Riccati ODE, leading to a solution.
  • #1
Ptopenny
3
0
the problem is (x+x^4)dy+y(y^3-x^3)dx=0
well I know that this is not a separable equation, homogenous equation or an exact equation...so i try to solve it by treating it as a non exact DE by finding out the integrating factor...but the both IF come out in term of x and y which involve 2 variables where by IF must only has one variable...
 
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  • #2
Hi !
The pattern of the ODE makes think to a change of variables : X=x^3 and Y=y^3, which leads to a Riccati ODE.
 

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  • #3
thank god...u really help me very much :D :D
 

1. What is a non-exact differential equation?

A non-exact differential equation is a type of differential equation where the total differential of the equation's solution cannot be expressed as the product of the equation's coefficients and a single differential term. In other words, the equation does not satisfy the condition of being an exact differential equation.

2. How is a non-exact differential equation different from an exact differential equation?

The main difference between a non-exact and an exact differential equation is that in a non-exact equation, the coefficients of the differential terms cannot be used to find the solution directly. Instead, additional integrating factors must be incorporated into the equation in order to solve it.

3. What is an integrating factor in a non-exact differential equation?

An integrating factor is a function that is multiplied to both sides of a non-exact differential equation in order to make it exact. It is typically a function of the independent variable in the equation, and when multiplied, it transforms the non-exact equation into an exact one, making it easier to solve.

4. How do you solve a non-exact differential equation?

To solve a non-exact differential equation, one must first identify it as non-exact and then find an appropriate integrating factor. The integrating factor is then multiplied to both sides of the equation, transforming it into an exact equation. From there, standard techniques for solving exact differential equations can be used, such as separation of variables or the method of integrating factors.

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

Non-exact differential equations are commonly used in many fields of science and engineering, such as physics, chemistry, and economics. They can be used to model a variety of physical phenomena, including chemical reactions, population growth, and heat transfer. They are also useful in analyzing and predicting behaviors of complex systems, such as weather patterns and stock market fluctuations.

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