How Can the Frobenius Method Be Used to Solve Complex ODEs?

In summary: So in summary, the conversation discusses the use of the Frobenius method to find a solution to a second order ODE with a regular singularity at x=0. The method involves finding a particular solution and then using it to find another independent solution. However, in this case, the second solution obtained is not analytic at x=0 and thus the Frobenius method cannot provide any information about it.
  • #1
Usagi
45
0
Ok here's a funny ODE to solve:

[tex]xy'' + (1-2x)y' + (x-1)y = 0[/tex]

clearly a straight forward power series substitution won't work here since we have a regular singularity at x = 0

so try the frobenius method by expanding around x = 0.

Assume [tex]y = \sum_{m=0}^{\infty} a_mx^{m+r}[/tex] is a solution where [tex]r[/tex] is some constant.

So we have [tex]y' = \sum_{m=0}^{\infty} (m+r)a_mx^{m+r-1}[/tex] and [tex]y'' = \sum_{m=0}^{\infty}(m+r)(m+r-1) a_mx^{m+r-2}[/tex]

Put this back in:

[tex]x \left(\sum_{m=0}^{\infty}(m+r)(m+r-1) a_mx^{m+r-2}\right) + (1-2x) \left(\sum_{m=0}^{\infty} (m+r)a_mx^{m+r-1}\right) + (x-1) \left(\sum_{m=0}^{\infty} a_mx^{m+r}\right) = 0[/tex]

after some algebra and stuff:

[tex]\sum_{m=0}^{\infty} (m+r)^2 a_m x^{m+r-1} - \sum_{m=0}^{\infty} [2(m+r)+1] a_m x^{m+r} + \sum_{m=0}^{\infty} a_m x^{m+r+1} = 0[/tex]

clearly lowest term is [tex]x^{r-1}[/tex] with it's coefficient as[tex] r^2a_0[/tex] hence [tex]r^2a_0 = 0[/tex]

Now [tex]a_0 \neq 0[/tex], so [tex]r^2 = 0 \implies r = 0[/tex]

Now we find the coefficients of the term [tex]x^s[/tex] where s is some constant, this gives:

[tex](s+1)^2a_{s+1} x^s - (2s+1) a_s x^s + a_{s-1} x^s = 0[/tex]

rearranging gives:

[tex]a_{s+1} = \frac{(2s+1)a_s - a_{s-1}}{(s+1)^2}[/tex] for s = 1, 2, etc

Thus we found a recurrence relationship with [tex]a_0[/tex] and [tex]a_1[/tex] as arbitrary initial values.

A bit of playing around quickly shows that:

[tex]a_2 = \frac{3a_1 - a_0}{4}[/tex]

[tex]a_3 = \frac{11a_1 - 5a_0}{36}[/tex]

[tex]a_4 = \frac{25a_1 - 13a_0}{288}[/tex]

Thus we have one of the solutions to be [tex]y = a_0 + a_1x + (\frac{3a_1 - a_0}{4})x^2 + (\frac{11a_1 - 5a_0}{36})x^3 + (\frac{25a_1 - 13a_0}{288})x^4 + ...[/tex]

However because a_0 and a_1 are arbitrary, let us pick... [tex]a_0 = a_1 = 1 [/tex], now magically we have:

[tex]y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}+ ... = e^x[/tex]

So [tex]y = e^x[/tex] is one of the basis for the general solution of this ode.

Now I was wondering, since [tex]a_0[/tex] and [tex]a_1[/tex] are arbitrary, then would ANY [tex]a_0[/tex] ([tex]\neq 0[/tex]) and [tex]a_1[/tex] work? Say [tex]a_0 = 4[/tex] and [tex]a_1 = 3[/tex] which then implies that there is an "infinite" number of different basis for the general solution of this ode?

Thanks
 
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  • #2
Usagi said:
Ok here's a funny ODE to solve:

[tex]xy'' + (1-2x)y' + (x-1)y = 0[/tex]

clearly a straight forward power series substitution won't work here since we have a regular singularity at x = 0

so try the frobenius method by expanding around x = 0.

Assume [tex]y = \sum_{m=0}^{\infty} a_mx^{m+r}[/tex] is a solution where [tex]r[/tex] is some constant.

So we have [tex]y' = \sum_{m=0}^{\infty} (m+r)a_mx^{m+r-1}[/tex] and [tex]y'' = \sum_{m=0}^{\infty}(m+r)(m+r-1) a_mx^{m+r-2}[/tex]

Put this back in:

[tex]x \left(\sum_{m=0}^{\infty}(m+r)(m+r-1) a_mx^{m+r-2}\right) + (1-2x) \left(\sum_{m=0}^{\infty} (m+r)a_mx^{m+r-1}\right) + (x-1) \left(\sum_{m=0}^{\infty} a_mx^{m+r}\right) = 0[/tex]

after some algebra and stuff:

[tex]\sum_{m=0}^{\infty} (m+r)^2 a_m x^{m+r-1} - \sum_{m=0}^{\infty} [2(m+r)+1] a_m x^{m+r} + \sum_{m=0}^{\infty} a_m x^{m+r+1} = 0[/tex]

clearly lowest term is [tex]x^{r-1}[/tex] with it's coefficient as[tex] r^2a_0[/tex] hence [tex]r^2a_0 = 0[/tex]

Now [tex]a_0 \neq 0[/tex], so [tex]r^2 = 0 \implies r = 0[/tex]

Now we find the coefficients of the term [tex]x^s[/tex] where s is some constant, this gives:

[tex](s+1)^2a_{s+1} x^s - (2s+1) a_s x^s + a_{s-1} x^s = 0[/tex]

rearranging gives:

[tex]a_{s+1} = \frac{(2s+1)a_s - a_{s-1}}{(s+1)^2}[/tex] for s = 1, 2, etc

Thus we found a recurrence relationship with [tex]a_0[/tex] and [tex]a_1[/tex] as arbitrary initial values.

A bit of playing around quickly shows that:

[tex]a_2 = \frac{3a_1 - a_0}{4}[/tex]

[tex]a_3 = \frac{11a_1 - 5a_0}{36}[/tex]

[tex]a_4 = \frac{25a_1 - 13a_0}{288}[/tex]

Thus we have one of the solutions to be [tex]y = a_0 + a_1x + (\frac{3a_1 - a_0}{4})x^2 + (\frac{11a_1 - 5a_0}{36})x^3 + (\frac{25a_1 - 13a_0}{288})x^4 + ...[/tex]

However because a_0 and a_1 are arbitrary, let us pick... [tex]a_0 = a_1 = 1 [/tex], now magically we have:

[tex]y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}+ ... = e^x[/tex]

So [tex]y = e^x[/tex] is one of the basis for the general solution of this ode.

Now I was wondering, since [tex]a_0[/tex] and [tex]a_1[/tex] are arbitrary, then would ANY [tex]a_0[/tex] ([tex]\neq 0[/tex]) and [tex]a_1[/tex] work? Say [tex]a_0 = 4[/tex] and [tex]a_1 = 3[/tex] which then implies that there is an "infinite" number of different basis for the general solution of this ode?

Thanks

In...

http://www.mathhelpboards.com/f17/another-second-order-non-homogeneous-ode-2237/#post10243

... it has been explained, given a second order ODE like...

$\displaystyle y^{\ ''} + p(x)\ y^{\ '} + q(x)\ y =0$ (1)

... a particular solution of which that we call v(x) is known, how to find another solution u(x) independent from v(x)...

$\displaystyle u= v\ \int \frac{e^{-\int p(x)\ dx}}{v^{2}}\ dx$ (2)

In Your case is $\displaystyle p(x)= \frac{1-2\ x}{x}$ and You found $v(x) = e^{x}$, so that is...

$\displaystyle e^{- \int p(x)\ dx} = \frac{e^{2\ x}}{x} \implies u(x) = e^{x}\ \int \frac{dx}{x} = e^{x}\ \ln x$ (2)

Now it is clear that u(x) is not analytic in x=0 and the Frobenoius method, applied in x=0, can't give any information about it... Kind regards $\chi$ $\sigma$
 
  • #3
ok thanks for that, but what if i picked a_0 = 1 and a_1 = 3?

Then the first solution would be an infinite sum and not e^x, how do i then find the second solution?
 
  • #4
chisigma said:
In...

http://www.mathhelpboards.com/f17/another-second-order-non-homogeneous-ode-2237/#post10243

... it has been explained, given a second order ODE like...

$\displaystyle y^{\ ''} + p(x)\ y^{\ '} + q(x)\ y =0$ (1)

... a particular solution of which that we call v(x) is known, how to find another solution u(x) independent from v(x)...

$\displaystyle u= v\ \int \frac{e^{-\int p(x)\ dx}}{v^{2}}\ dx$ (2)

In Your case is $\displaystyle p(x)= \frac{1-2\ x}{x}$ and You found $v(x) = e^{x}$, so that is...

$\displaystyle e^{- \int p(x)\ dx} = \frac{e^{2\ x}}{x} \implies u(x) = e^{x}\ \int \frac{dx}{x} = e^{x}\ \ln x$ (2)

Now it is clear that u(x) is not analytic in x=0 and the Frobenoius method, applied in x=0, can't give any information about it... Kind regards $\chi$ $\sigma$

... and for supply an as complete as possible answer, the general solution of the proposed ODE is...

$\displaystyle y(x)= c_{1}\ e^{x} + c_{2}\ e^{x}\ \ln x$ (1)

... if You intend to set the coefficients $a_{0}$ and $a_{1}$, necessarly must be $a_{0}=a_{1}=a$, and the reason of that is evident observing (1)...

Kind regards

$\chi$ $\sigma$
 
  • #5
Thanks, however I was told that as long as we don't set a_0 to be 0, then a_0 and a_1 can be anything and it will still be a perfectly valid solution for the ode? It's just that it makes finding the second solution harder than if we set a_0 = a_1?
 
  • #6
Usagi said:
Thanks, however I was told that as long as we don't set a_0 to be 0, then a_0 and a_1 can be anything and it will still be a perfectly valid solution for the ode? It's just that it makes finding the second solution harder than if we set a_0 = a_1?

If y(x) is the solution You are searching and is $\displaystyle y(x)= \sum_{n=0}^{\infty} a_{n}\ x^{n}$, then it must be necessarly $a_{0}= y(0)$ and $a_{1}= y^{\ '}(0)$... right?... but if the general solution is...

$y(x)= c_{1}\ e^{x} + c_{2}\ e^{x}\ \ln x$ (1)

... y(x) and its derivative don't exist in x=0 unless is $c_{2}=0$, but if so, then il must be necessarly $y(0)=y^{\ '}(0)$...

The conclusion is : imposing $a_{0} \ne a{1}$ is a contradiction...

Kind regards

$\chi$ $\sigma$
 
  • #7
thanks that makes sense!

What about this one:

[tex]y'' + (x-1)y = 0[/tex]

[tex]\sum_{m=0}^{\infty} (m+r)(m+r-1)a_m x^{m+r-2} + (x-1) \sum_{m=0}^{\infty} a_m x^{m+r} = 0[/tex]

[tex]\sum_{m=0}^{\infty} (m+r)(m+r-1)a_m x^{m+r-2} + \sum_{m=0}^{\infty} a_m x^{m+r+1} - \sum_{m=0}^{\infty} a_m x^{m+r} = 0[/tex]

so r(r-1) = 0 -> r= 0, 1

Consider r = 1:

[tex]\sum_{m=0}^{\infty} (m+1)m a_m x^{m-1} + \sum_{m=0}^{\infty} a_m x^{m+2} - \sum_{m=0}^{\infty} a_m x^{m+1} = 0[/tex]

Let us find the coefficient of x^s

[tex](s+2)(s+1)a_{s+1} + a_{s-2} - a_{s-1} = 0[/tex]

so [tex]a_{s+1} = \frac{a_{s-1} - a_{s-2}}{(s+2)(s+1)}[/tex] for s = 2, 3, 4, ...

so then how do i know what values to set a_0, a_1 and a_2?

and also how would i find the other solution for the basis?
 
  • #8
Usagi said:
thanks that makes sense!

What about this one:

[tex]y'' + (x-1)y = 0[/tex]

[tex]\sum_{m=0}^{\infty} (m+r)(m+r-1)a_m x^{m+r-2} + (x-1) \sum_{m=0}^{\infty} a_m x^{m+r} = 0[/tex]

[tex]\sum_{m=0}^{\infty} (m+r)(m+r-1)a_m x^{m+r-2} + \sum_{m=0}^{\infty} a_m x^{m+r+1} - \sum_{m=0}^{\infty} a_m x^{m+r} = 0[/tex]

so r(r-1) = 0 -> r= 0, 1

Consider r = 1:

[tex]\sum_{m=0}^{\infty} (m+1)m a_m x^{m-1} + \sum_{m=0}^{\infty} a_m x^{m+2} - \sum_{m=0}^{\infty} a_m x^{m+1} = 0[/tex]

Let us find the coefficient of x^s

[tex](s+2)(s+1)a_{s+1} + a_{s-2} - a_{s-1} = 0[/tex]

so [tex]a_{s+1} = \frac{a_{s-1} - a_{s-2}}{(s+2)(s+1)}[/tex] for s = 2, 3, 4, ...

so then how do i know what values to set a_0, a_1 and a_2?

and also how would i find the other solution for the basis?

Althogh it may seem 'simple' the second order linear ODE...

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

... is not very comfortable and its general solution is...

$\displaystyle y= c_{1}\ \text{Ai} (1-x) + c_{2}\ \text{Bi} (1-x)$ (2)... where Ai(*) and Bi (*) are the so called 'Airy Functions'...Airy Functions -- from Wolfram MathWorld

... and may be that the Frobenius approach is not the best way to attack (1)...

Kind regards

$\chi$ $\sigma$
 

FAQ: How Can the Frobenius Method Be Used to Solve Complex ODEs?

What is the Frobenius method?

The Frobenius method is a mathematical technique used to solve ordinary differential equations. It involves assuming a solution in the form of a power series and then finding the coefficients that satisfy the differential equation.

When is the Frobenius method used?

The Frobenius method is typically used to solve differential equations that cannot be solved using other methods, such as separation of variables or substitution. It is commonly used in physics and engineering applications.

How does the Frobenius method work?

The Frobenius method involves finding a power series solution to a differential equation, substituting it into the equation, and solving for the coefficients. The resulting series will be a solution to the differential equation if the coefficients satisfy certain conditions.

What are the limitations of the Frobenius method?

The Frobenius method can only be used for linear differential equations with regular singular points. It also may not always result in a finite series solution, and the convergence of the series may be limited to a specific range of values.

Can the Frobenius method be used for partial differential equations?

No, the Frobenius method is only applicable to ordinary differential equations. For partial differential equations, other methods such as separation of variables or the method of characteristics must be used.

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