Problem with first-order nonlinear ordinary differential equation

In summary, the conversation is about a person trying to find a solution for a given problem using the exact equation method. They identify the M and N variables and determine that the equation is not exact. They then try to find an integrating factor, but are unable to do so because the equation is a function of both x and y. They ask for help in solving the DE but are informed that not all ODEs have an easy integrating factor.
  • #1
bobey
32
0
i have problem to find the solution for : (3x3y+2xy+y3)+(x2+y2)dy/dx=0

i have tried the exact equation method :

(3x3y+2xy+y3)dx+(x2+y2)dy=0

thus M(x,y)=(3x3y+2xy+y3)

and N(x,y)= (x2+y2)

then deltaM/deltay=3x3+2x+3y2
and deltaN/deltax=2x

Since deltaM/deltay does not equal to deltaN/deltax, this imply that the equation is not exact

thus, finding/searching for integrating factor :

1. 1/N(deltaM/deltay-deltaN/deltax)=(3x3+3y3)/(3x3y+2xy+y3)

y cannot be eliminated . thus, this is a function of both x and y, not just x

2. 1/M(deltaN/deltax-deltaM/deltay)=(3x3+3y2)/(x2+y2)

x cannot be eliminated . thus, this is a function of both x and y, not just y


thus i cannot find the integrating factor in order to solve the DE. where I'm gone wrong? can somebody point it out? i guess may be in algebra...
 
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  • #2
You are on right direction. But not all of ODEs have an easy integrating factor.
 
  • #3
is there anyway for me to solve the de? please help me...
 

1. What is a first-order nonlinear ordinary differential equation?

A first-order nonlinear ordinary differential equation is an equation that relates a function and its derivative, where the function is not a linear function. This means that the equation cannot be written in the form of y = mx + b, where m and b are constants.

2. What makes solving first-order nonlinear ordinary differential equations challenging?

Solving first-order nonlinear ordinary differential equations can be challenging because there is no general method for solving them. The solution often involves finding an appropriate substitution or transformation to turn the equation into a more manageable form.

3. Can first-order nonlinear ordinary differential equations have multiple solutions?

Yes, first-order nonlinear ordinary differential equations can have multiple solutions. This is because the derivative of a function can take on different values at different points, resulting in multiple possible solutions to the equation.

4. How are first-order nonlinear ordinary differential equations used in science?

First-order nonlinear ordinary differential equations are used to model a wide range of natural phenomena in science. They can be used to describe the behavior of physical systems, population growth, chemical reactions, and many other processes.

5. Are there any real-world applications of first-order nonlinear ordinary differential equations?

Yes, there are many real-world applications of first-order nonlinear ordinary differential equations. These equations are used in fields such as physics, chemistry, biology, economics, and engineering to model and understand complex systems and predict future behavior.

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