Solving a Second Order Non-Linear PDE with Undetermined Coefficients

• binbagsss
In summary, the conversation discusses a non-linear second order PDE with two different simplified versions for easy solving. However, the given equation does not fit into either category and may require an integrating factor for solving. One suggestion is to experiment with polynomial or rational function forms for y.
binbagsss

Homework Statement

##\frac{d^2y}{dx^2}=2xy\frac{dy}{dx}##

Homework Equations

This is second order non-linear pde of the 'form' ## f(y'',y',y,x) ## .

I have read that there are 2 simplified versions of a second order non-linear pde that can be solved easily and these are 1) when there is no y term 2) when there is no x term

The Attempt at a Solution

[/B]
The above does not fit into these two categories, and as such I think it has no general sort of procedure or way to solve, but it is possible for an integrating factor to come for mind for this particular one? (i just have no idea what this could be !)

Many thanks

binbagsss said:

Homework Statement

##\frac{d^2y}{dx^2}=2xy\frac{dy}{dx}##

Homework Equations

This is second order non-linear pde of the 'form' ## f(y'',y',y,x) ## .

I have read that there are 2 simplified versions of a second order non-linear pde that can be solved easily and these are 1) when there is no y term 2) when there is no x term

The Attempt at a Solution

[/B]
The above does not fit into these two categories, and as such I think it has no general sort of procedure or way to solve, but it is possible for an integrating factor to come for mind for this particular one? (i just have no idea what this could be !)

Many thanks
Putting the equation in this form
##\frac{y''}{yy'}=2x##
suggests that ##y## is a polynomial or a rational function. Experiment with these forms using undetermined coefficients and you will quickly find a solution.

tnich said:
Putting the equation in this form
##\frac{y''}{yy'}=2x##
suggests that ##y## is a polynomial or a rational function. Experiment with these forms using undetermined coefficients and you will quickly find a solution.

This type of suggestion would not work for the simpler similar case ##\frac{y'}{y} = 2x##.

Anyway, if ##y## is a polynomial of degree ##n##, the right-hand-side ##2 x y y'## is a polynomial of degree ##2n## while the left-hand-side ##y''## is a polynomial of degee ##n-2##. That requires ##2n = n-2##, hence ##n = -2##. That suggests a solution of the form ##1/x^2##!

1. What is a 2nd order non-linear PDE?

A 2nd order non-linear PDE (partial differential equation) is a mathematical equation that involves multiple variables and their partial derivatives up to the second order. It is non-linear because the equation does not follow the standard form of a linear equation, where the variables are raised to the first power only.

2. What is the difference between a 2nd order non-linear PDE and a 1st order PDE?

The main difference between a 2nd order non-linear PDE and a 1st order PDE is the number of variables involved and the highest order of partial derivatives present in the equation. A 1st order PDE only involves first-order partial derivatives, while a 2nd order non-linear PDE involves up to second-order partial derivatives.

3. How are 2nd order non-linear PDEs used in science?

2nd order non-linear PDEs are used in various fields of science, such as physics, biology, and engineering, to model and describe complex systems. They are particularly useful in studying systems that exhibit non-linear behavior, such as chaotic systems or systems with feedback loops.

4. What are some common methods for solving 2nd order non-linear PDEs?

Some common methods for solving 2nd order non-linear PDEs include separation of variables, method of characteristics, and numerical methods such as finite difference or finite element methods. The choice of method depends on the specific equation and boundary conditions.

5. Are there any real-world applications of 2nd order non-linear PDEs?

Yes, there are many real-world applications of 2nd order non-linear PDEs. Some examples include modeling heat transfer in materials, predicting the behavior of fluids in pipes, and studying the dynamics of chemical reactions. They are also used in the development of computer simulations for various systems and processes.

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