Non-linear differential equation

In summary, the system $x'=y+xf(r^2)$ and $y'=-x+yf(r^2)$, where $r^2=x^2+y^2$, has a solution of $\frac{dr^2}{dt}=2r^2f(r^2)$ by using the chain rule and the fact that $rr'=xx'+yy'$. Assuming $f(r^2)$ has N zeroes, the system has only one fixed point at (0,0) and the stability of this fixed point is justified by ${\theta}'=-1$. There may be periodic solutions, but this is not clear.
  • #1
Poirot1
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0
system is $x'=y+xf(r^2)$ and $y'=-x+yf(r^2)$. where $r^2=x^2+y^2$

(i) prove that $\frac{dr^2}/{dt}=2r^2f(r^2)$. My solution ( I won't write out details): use chain rule and the fact that rr'=xx'+yy'.

(ii)assume $f(r^2)$ has N zeroes. determine the number of fixed points and periodic solutions the system has and write about the stability of fixed points.

This one I think I did the first (I will give details) but can't do the others

Solution: $r^2{\theta}'=xy'-yx'$and if (x,y) is fixed point, then xy'-yx'=0. If you work this out you get fixed point implies -r^2=0 so x=y=0 is only fixed point. In the solution however it has a different justification, namely that ${\theta}'=-1$ but I don't understand how this means (0,0) is only fixed point.
 
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  • #2
You already know that x'= y+ xf(r^2) and y'= -x+ yf(r^2) so that xy'- yx'= x(-x+ yf(r^2))- y(y+ xf(r^2)= -x^2+ xyf(r^2)- (y^2+ xyf(r^2)= -(x^2+y^2)= -r^2= 0 if and only if r= 0.
 
  • #3
ok what about the question about periodic solutions
 

Related to Non-linear differential equation

1. What is a non-linear differential equation?

A non-linear differential equation is a type of mathematical equation that involves derivatives of a dependent variable with respect to one or more independent variables. The equation is considered non-linear if the dependent variable appears with an exponent greater than one or is multiplied by another variable.

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

A linear differential equation is one in which the dependent variable and its derivatives appear in a linear manner, meaning they are only raised to the first power and are not multiplied by other variables. Non-linear differential equations, on the other hand, can have terms with exponents greater than one and can involve multiplication of variables.

3. What are some applications of non-linear differential equations in science?

Non-linear differential equations are commonly used in physics, engineering, and other scientific fields to model complex systems. They are particularly useful in studying phenomena such as chaotic behavior, population dynamics, and chemical reactions.

4. Are there analytical solutions to non-linear differential equations?

In most cases, there are no analytical solutions to non-linear differential equations. Instead, numerical methods and computer simulations are often used to approximate solutions. However, some simpler non-linear equations may have analytical solutions.

5. How do you solve a non-linear differential equation?

The method for solving a non-linear differential equation depends on its specific form and the tools available. Some may be solved using separation of variables, substitution, or other algebraic techniques. Others may require numerical methods or computer simulations. In general, non-linear differential equations are more difficult to solve than linear ones and may not have a single, exact solution.

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