Solve the ODE y'' + (3x)/(1+x^2)y' + 1/(1+x^2)y = 0

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In summary: Remember you have two series, each with its own initial condition. So you will end up with two recurence relations, one for the even terms and one for the odd terms. From there you can find the general solution for each series. Good luck!In summary, in order to determine the fundamental set of solutions of the given differential equation in the form of a power series, one must first multiply the equation by (1+x^2) and then make a change of index on the summation. After grouping the coefficients, a recurrence relation can be found to solve for the general solution of each series.
  • #1
Treadstone 71
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I need to determine the fundamental set of solutions of

[tex]y''+\frac{3x}{1+x^2}y'+\frac{1}{1+x^2}y=0[/tex]

in form of a power series centered around [tex]x_0=0[/tex].

When I expanded the y's in power series, I am unable to bring the 1/1+x^2 inside the infinite sum. Can anyone help?
 
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  • #2
Try multiplying the whole equation by [tex] 1 + x^2 [/tex].
 
  • #3
Treadstone 71 said:
I need to determine the fundamental set of solutions of
[tex]y''+\frac{3x}{1+x^2}y'+\frac{1}{1+x^2}y=0[/tex]
in form of a power series centered around [tex]x_0=0[/tex].
When I expanded the y's in power series, I am unable to bring the 1/1+x^2 inside the infinite sum. Can anyone help?

Multiply throughout by [itex](1+x^2)[/itex] first to get:

[tex](1+x^2)y^{''}+3xy^{'}+y=0[/tex]

So wouldn't that first term just be:

[tex]\sum n(n-1)a_n x^{n-2}+\sum n(n-1)a_nx^n[/tex]

right?
 
  • #4
There is much to this problem after you get passed your issue, however; I will restrain my response to your question. First off, multiply everything by 1+x^2. Now after subsituting your power series expression for y, y', and y'' you will be left with only 2 powers of x: x^n-2 and x^n. I imagine your question begins here although now in different form. You must make a change of index on the summation containing x^n-2, try m=n-2. After this substitution you will have an expression with a sum with x^m and one with x^n. Since "m" is a dummy index let m=n. It seems circular but it is correct and required. Now you will have an expression in which the only powers of x that appear are x^n. Grouping the coefficients and setting them equal to zero is your next step which I will let you do on your own. It is interesting to note that in order to do this problem you are switching the "shift" of the powers of x to the coefficients an. Not only is this helpful, but it is mandatory when solving the original DE since you must have different an's from which to find your recurrence relations.
 
  • #5
If I make a substitution m=n-2, at some point I'll have a term in the form of:

[tex]\sum_{n=0}(n+2)(n+1)a_{n+2}x^{n+2}[/tex]

Are you saying that at this point I can reset the index of x to n and leave a_n the way it is?
 
  • #6
That is not what you will have. The coefficients look correct but its not x^(n+2) it is x^n. Recall: you had x^(n-2), you let m=n-2, therefore
x^((m+2)-2)=x^m with the coefficients you have. Then subsititute back and let m=n.
 
  • #7
Ok, got it, I think:

[tex]\sum_{n=0}^{\infty}x^n[a_{n+2}(n+2)(n+1)+a_n[n(n-1)+3n+1]][/tex]

This implies that

[tex]a_{n+2}(n+2)(n+1)+a_n(n+1)^2=0[/tex]
 
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  • #8
yes, now find your recurence relation and your done.
 

1. What is an ODE?

An ODE, or ordinary differential equation, is a mathematical equation that describes the relationship between a function and its derivatives. In this case, the function is y and its derivatives are y' and y''.

2. What does the notation y'' + (3x)/(1+x^2)y' + 1/(1+x^2)y = 0 mean?

The notation y'' + (3x)/(1+x^2)y' + 1/(1+x^2)y = 0 is a shorthand way of writing the ODE. The double prime (") represents the second derivative of y, and the prime (') represents the first derivative. The numbers and expressions in parentheses indicate coefficients in front of the derivatives, and the entire equation equals 0.

3. Why does this ODE have a solution?

This ODE has a solution because it is a linear homogeneous equation with constant coefficients. This means that the equation can be rearranged into a specific form that allows for a solution to be found using mathematical techniques.

4. How do you solve this ODE?

This ODE can be solved using various techniques such as separation of variables, variation of parameters, or integrating factors. The specific method used depends on the form of the equation and the skills of the person solving it.

5. What are the applications of solving this ODE?

Solving this ODE can have various applications in the fields of science, engineering, and mathematics. It can be used to model and predict the behavior of physical systems, such as in mechanics or fluid dynamics. It can also be used in statistics and economics to model relationships between variables.

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