Exploring Singular Points of a DE: w''+z*w'+kw=0

In summary, a singular point in a differential equation is a point where the solution does not exist, becomes infinite or undefined, or has a drastic change in behavior. These points can be identified by setting the coefficients of the highest order derivative term to zero and solving for the independent variable. They can be classified as regular or irregular, with different properties for each type. Exploring singular points can help understand the solution's behavior and identify important features, and techniques such as power series solutions, Frobenius method, and phase plane analysis can be used to analyze them.
  • #1
Ice Vox
3
0
Hi,
I'm asked to find and classify the singular points of a function w(z) in the differential equation:

w''+z*w'+kw=0 where k is some unknown constant.

The only singular point I notice is \(\displaystyle z=\infty\). Is that right?

I did a transformation x=1/z and examined the singular point at x=0 and found that the limit as \(\displaystyle x\to0\) gives \(\displaystyle 2-1/x^2\) which blows up, meaning the singular point is irregular. Does that mean that the singular point z=infinity is irregular also?

It also asks me to find the first term of the asymptotic solution as \(\displaystyle z\to\infty\) for each of the two solutions. Does this mean just put it in a power series solution \(\displaystyle w=\sum_{n=0}^\infty a_nz^n\) and see what happens?

Thanks!
 
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  • #2
CWolf said:
Hi,
I'm asked to find and classify the singular points of a function w(z) in the differential equation:

w''+z*w'+kw=0 where k is some unknown constant.

The only singular point I notice is \(\displaystyle z=\infty\). Is that right?

I did a transformation x=1/z and examined the singular point at x=0 and found that the limit as \(\displaystyle x\to0\) gives \(\displaystyle 2-1/x^2\) which blows up, meaning the singular point is irregular. Does that mean that the singular point z=infinity is irregular also?

It also asks me to find the first term of the asymptotic solution as \(\displaystyle z\to\infty\) for each of the two solutions. Does this mean just put it in a power series solution \(\displaystyle w=\sum_{n=0}^\infty a_nz^n\) and see what happens?

Thanks!

Wellcome on MHB CWolf!...

In general a second order linear homogeneous DE can be written as...

$\displaystyle y^{\ ''} + f_{1} (x)\ y^{\ '} + f_{2} (x)\ y =0\ (1)$

We have the following possible cases...

a) if $f_{1} (x)$ and $f_{2} (x)$ are both analytic in x=a, then x=a is an ordinary point...

b) if $f_{1} (x)$ has a pole up to order 1 and $f_{2}(x)$ a pole up to order 2 in x=a, then x=a is a regular singular point...

c) in any other case x=a is a singular point...

In Your case is $f_{1}(x) = x$ and $f_{2} = k$, both analytic in $\mathbb R$, so that there are no singular points...

Kind regards

$\chi$ $\sigma$
 

1. What is a singular point in a differential equation?

A singular point in a differential equation is a point where the solution does not exist, or where the solution becomes infinite or undefined. It can also be described as a point where the behavior of the solution changes drastically.

2. How are singular points identified in a differential equation?

Singular points can be identified by setting the coefficients of the highest order derivative term to zero and solving for the independent variable. This will give the values of the independent variable at which the singular points occur.

3. How are singular points classified in a differential equation?

Singular points can be classified as regular or irregular. A regular singular point has a finite radius of convergence for its power series solution, while an irregular singular point has an infinite radius of convergence.

4. What is the significance of exploring singular points in a differential equation?

Exploring singular points can help in understanding the behavior of the solution of a differential equation, especially near the singular points. It can also help in identifying important features of the solution, such as critical points and stability.

5. What are some techniques used to analyze singular points in a differential equation?

Some techniques used to analyze singular points include power series solutions, Frobenius method, and phase plane analysis. These methods can help determine the nature of the singular points and their effect on the overall solution of the differential equation.

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