Non-linear second-order ODE to Fuchsian equation

In summary, the given equation is linearized by substituting z=\sqrt y and t=\frac{1}{w}. After substitutions, a final equation is obtained which can be solved to find the general solution. However, to obtain a more general solution, k can be allowed to be a function of t or w using variation of constants method.
  • #1

Homework Statement



[tex]z\frac{d^2z}{dw^2}+\left(\frac{dz}{dw}\right)^2+\frac{\left(2w^2-1\right)}{w^3}z\frac{dz}{dw}+\frac{z^2}{2w^4}=0[/tex]

(a) Use [itex]z=\sqrt y[/itex] to linearize the equation.
(b) Use [itex]t=\frac{1}{w}[/itex] to make singularities regular.
(c) Solve the equation.
(d) Is the last equation obtained a Fuchsian equation?
(e) Discuss what you learn about [itex]z(w)[/itex] with this strategy.

Homework Equations



Source equation:
[tex]z\frac{d^2z}{dw^2}+\left(\frac{dz}{dw}\right)^2+\frac{\left(2w^2-1\right)}{w^3}z\frac{dz}{dw}+\frac{z^2}{2w^4}=0[/tex]

Substitutions:
[tex]z=\sqrt y[/tex], [tex]t=\frac{1}{w}[/tex]

The Attempt at a Solution


[/B]
First, I did [itex]z=\sqrt y[/itex], such that [itex]\frac{dz}{dw}=\frac{1}{2\sqrt y}\frac{dy}{dw}[/itex], and [itex]\frac{d^2z}{dw^2}=\frac{1}{2\sqrt y}\frac{d^2y}{dw^2}-\frac{1}{4y^{3/2}}\left(\frac{dy}{dw}\right)^2[/itex]. When substituted in the equation above, I got

[tex]\frac{d^2y}{dw^2}+\frac{(2w^2-1)}{w^3}\frac{dy}{dw}+\frac{1}{w^4}y=0[/tex].

Then I tried [itex]t=\frac{1}{w}[/itex], such that [itex]\frac{dy}{dw}=-t^2\frac{dy}{dt}[/itex], and [itex]\frac{d^2y}{dw^2}=t^4\frac{d^2y}{dt^2}+2t^3\frac{dy}{dt}[/itex].

From here, I ended at

[tex]\frac{d^2y}{dt^2}+t\frac{dy}{dt}+y=0[/tex].

This doesn't look right. For it could be rewritten as [itex]\frac{d}{dt}\left(\frac{dy}{dt}+ty\right)=0 \Rightarrow y(x)=ke^{(-x^2/2)}[/itex], and going back the substitutions I'd have [itex]z(w)=\sqrt{ke^{-(1/w)^2/2}}[/itex], which when substituted in the original equation yields only the trivial solution [itex]k=0[/itex].

Where did I go wrong?
I didn't type all my calculations to not clutter. Please ask if necessary.
 
Physics news on Phys.org
  • #2
Your solution is not general it should have two constants.
It does work for all k check it again, or better yet check the general solution.
One possible method to add a constant is variation of a constant.
Allow k to be a function of t or w.
 

1. What is a non-linear second-order ODE?

A non-linear second-order ODE (ordinary differential equation) is a mathematical equation that describes the relationship between a function and its derivatives up to the second order. Non-linear means that the equation does not have a linear relationship between the function and its derivatives.

2. What is a Fuchsian equation?

A Fuchsian equation is a type of second-order ODE that can be written in a certain standard form. It has singularities at a finite number of points, and the coefficients of the equation are rational functions of the independent variable.

3. How is a non-linear second-order ODE converted to a Fuchsian equation?

To convert a non-linear second-order ODE to a Fuchsian equation, we can use a transformation called the Fuchsian transformation. This transformation involves changing the dependent and independent variables and introducing new parameters to the equation to make it fit the standard form of a Fuchsian equation.

4. What are the applications of Fuchsian equations?

Fuchsian equations have applications in various fields, such as physics, engineering, and mathematics. They are particularly useful in studying problems with singularities, such as in celestial mechanics, fluid dynamics, and quantum mechanics.

5. Are there any techniques for solving non-linear second-order ODEs to Fuchsian equations?

Yes, there are several techniques for solving non-linear second-order ODEs to Fuchsian equations, including the Fuchsian method, the power series method, and the Frobenius method. These methods involve finding a series solution for the equation and then using boundary conditions to determine the coefficients of the series.

Suggested for: Non-linear second-order ODE to Fuchsian equation

Replies
4
Views
343
Replies
2
Views
802
Replies
5
Views
94
Replies
1
Views
434
Replies
4
Views
582
Replies
2
Views
247
Back
Top