How can I find two linearly independent solutions for 4xy''+2y'+y=0?

In summary, the question is asking for help on finding two linearly independent solutions for 4xy+2y=0. The user has tried almost everything but has not been successful. They need to assume a solution and find the derivatives. Once they have the derivatives, they can substitute them into the equation and solve for y' and y".
  • #1
rapwaydown
10
0
the question is
" find two linearly independent of frebenius series solutions for 4xy''+2y'+y=0"

I try almost everything to slove this, but could't figure it
any help is appercaited

thank you
 
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  • #2
Well, what have you done on it? You say you've tried "almost everything". Okay, what have you tried?
 
  • #3
HallsofIvy said:
Well, what have you done on it? You say you've tried "almost everything". Okay, what have you tried?


well, i divided the whole thing by 4x, trying to get it to the general format, but it didnt work out.
 
  • #4
In other words, you really haven't done anything!

Start by writing
[tex]y= \sum_{n=0}^\infty a_n x^{n+c}[/tex]
Find y' and y" from that and put them into the equation. What do you get?
 
  • #5
HallsofIvy said:
In other words, you really haven't done anything!

Start by writing
[tex]y= \sum_{n=0}^\infty a_n x^{n+c}[/tex]
Find y' and y" from that and put them into the equation. What do you get?

i don't know how to find y' and y''
im slef learning def.Q
thanks for the reply tho.
 
  • #6
HallsofIvy said:
In other words, you really haven't done anything!

Start by writing
[tex]y= \sum_{n=0}^\infty a_n x^{n+c}[/tex]
Find y' and y" from that and put them into the equation. What do you get?

by the way, i don't think y' and y'' is needed here
i divided the whole thing by 4x, then slove for the indicial,
which are r=0,-.5
but i don't know what to do from this point on,
i think i need to find the equation for cn, but don't know how.
can you help
 
  • #7
You have to assume a solution as you have been advised
You don't need to divide by 4x.
When you assume the solution you will find d second and first derivatives of y, you will then subtitute into the equation.
I solved it and that's the way it goes.
 
  • #8
Though not the question it is interesting to know that this DE has a solution in closed form. One can find this by substituting [itex]x=t^2[/itex] in the equation. A very simple DE will appear and can be solved directly. Doing the inverse substituting on this solution gives the result of the original DE. This solution can then be compared to the series solution.
 
  • #9
rapwaydown said:
i don't know how to find y' and y''
im slef learning def.Q
thanks for the reply tho.
It's probably not a good idea to try to learn differential equations, by your self or not, if you do not know Calculus!

Are you seriously saying that you do not know how to find the derivatives of xn?

And, to even attempt a problem like this you should have had enough Calculus to know that a power series is "term by term" differentiable inside its radius of convergence.
 

Related to How can I find two linearly independent solutions for 4xy''+2y'+y=0?

1. What is a Frobenius series?

A Frobenius series is a type of power series used in mathematics to approximate solutions to differential equations. It was first introduced by the mathematician Ferdinand Georg Frobenius in the 1870s.

2. How is a Frobenius series different from a Taylor series?

A Taylor series is used to approximate functions near a single point, while a Frobenius series can approximate solutions to differential equations with a singularity at the point of expansion. This makes it more versatile for solving a wider range of problems.

3. When is a Frobenius series used?

A Frobenius series is typically used when the coefficients of a differential equation are not constant or when the equation has a singularity at the point of expansion. It is commonly used in physics, engineering, and other fields to approximate solutions to differential equations.

4. What is the process for finding a Frobenius series?

The process for finding a Frobenius series involves substituting the series into the given differential equation, solving for the coefficients, and then checking for convergence. It can be a lengthy process, but it is a powerful tool for solving differential equations with complicated coefficients or singularities.

5. Are there any limitations to using a Frobenius series?

While a Frobenius series can be a useful tool for solving differential equations, it may not always provide an exact solution. In some cases, the series may only provide an approximation, and further methods may be needed to find an exact solution. Additionally, the series may not converge for certain values of the independent variable, making it limited in its applicability.

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