# Insane differential equation help?

• itsthewoo
In summary, the conversation discusses a complicated equation and the process of finding its solution. The solution is found using an advanced technique called Lie point transformations and an integrating factor. The final solution is confirmed to be the same as the one given by other mathematical software. A book recommendation is also provided for further information.
itsthewoo
The equation is as follows:

2x * (dy/dx) + y = y2 * sqrt(x - (x2 * y2))

Any ideas how to solve it or begin solving it?

Hmm maple gives a fairly straightforward answer maybe a subst. u = xy, just a guess

Hello itsthewoo,

I finally found the solution. It kept running in my head and I couldn't put it aside, so here it is. First of all I used an advanced technique called Lie point transformations to find the answer. It comes down for this equation to find an integrating factor so the equation becomes exact. Writing the equation as:

$$\left(2x\right)\cdot dy-\left(-y+y^2\sqrt{x-x^2y^2}\right)\cdot dx=0$$

The integrating factor is the following:

$$M=\frac{1}{xy^2\sqrt{x-x^2y^2}}$$

The new (exact) equation is now:

$$\left(\frac{2x}{xy^2\sqrt{x-x^2y^2}}\right)\cdot dy+ \left(\frac{1}{xy\sqrt{x-x^2y^2}}-\frac{1}{x}\right)\cdot dx=0$$

Which has as solution:

$$x=Ke^{-\frac{2\sqrt{x-x^2y^2}}{xy}}$$

This is the same solution as maxima (like maple or mathematica) gives. If any more info is necessary, I would recommend the following book:

"Ordinary differential equations, an elementary text-book with an introduction
to Lie's theory of the group of one parameter." written by James Morris Page

Hope this helps,

coomast

$$u = x y^2$$ will separate your equation.

Hello Mathwebster,

I considered the following one-parameter group:

$$Uf=ax\frac{\partial f}{\partial x}+by\frac{\partial f}{\partial y}+(b-a)y'\frac{\partial f}{\partial y'}$$

And found the unknown numbers to bea=-1 and b=1/2. From this it is possible to find the integrating factor, but also the transformation you gave. It works indeed. I did not think of doing it like that although I should have . Many thanks for the hint.

coomast

## 1. What is an insane differential equation?

An insane differential equation is a type of mathematical equation that involves derivatives and nonlinear functions. These equations can be extremely complex and difficult to solve, often requiring advanced mathematical techniques and computer simulations.

## 2. Why do we need help with insane differential equations?

Insane differential equations are important in many areas of science and engineering, but they can be challenging to solve. As a result, many researchers and scientists may need assistance in understanding and solving these equations in order to make progress in their work.

## 3. How can I approach solving an insane differential equation?

The best approach to solving an insane differential equation depends on the specific equation and its complexity. However, some common techniques include separation of variables, substitution, and using numerical methods or computer simulations.

## 4. Are there any resources available for help with insane differential equations?

Yes, there are many resources available for help with insane differential equations. These include textbooks, online tutorials and courses, and consulting with experts in the field. Additionally, there are many software programs and online tools that can assist with solving these equations.

## 5. Can I use insane differential equations in my research or work?

Yes, insane differential equations are commonly used in various fields of science and engineering, such as physics, chemistry, and engineering. They can be used to model complex systems and make predictions about their behavior, making them an important tool for researchers and scientists.

Replies
1
Views
1K
Replies
7
Views
3K
Replies
1
Views
2K
Replies
2
Views
2K
Replies
14
Views
4K
Replies
1
Views
2K
Replies
3
Views
2K
Replies
52
Views
2K
Replies
1
Views
2K
Replies
1
Views
2K